____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \sin ^k(\lambda x) + b \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (a*Sin[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {\sin ^2(\lambda x)^{-\frac {k}{2}-\frac {1}{2}} \left (-a \cos (\lambda x) \sin ^{k+1}(\lambda x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-k}{2},\frac {3}{2},\cos ^2(\lambda x)\right )+b \lambda x \sin ^2(\lambda x)^{\frac {k}{2}+\frac {1}{2}}-\lambda y \sin ^2(\lambda x)^{\frac {k}{2}+\frac {1}{2}}\right )}{\lambda }\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(a*sin(lambda*x)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx-\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+y \right ) \] contains unresolved integral
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \sin ^k(\lambda y) + b \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (a*Sin[lambda*y]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \frac {1}{a \sin ^k(\lambda K[1])+b} \, dK[1]-x\right )\right \}\right \} \] contains unresolved integral
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(a*sin(lambda*y)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \sin \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \] contains unresolved integral
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \sin ^k(\lambda y) \sin ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + a*Sin[lambda*x]^k*Sin[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1()\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+a*sin(lambda*x)^k*sin(mu*y)^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \sin \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \] contains unresolved integral
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \sin ^k(x+\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + a*Sin[x + lambda*y]^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+a*sin(x+lambda*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {y\lambda +x}{\lambda }}}\! \left ( 1+a \left ( \sin \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}\lambda \right ) ^{-1}{d{\it \_a}}\lambda +x \right ) \] contains unresolved integral
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left (y^2-a^2 + a \lambda \sin (\lambda x)+a^2 \sin ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (y^2 - a^2 + a*lambda*Sin[lambda*x] + a^2*Sin[lambda*x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(y^2-a^2 + a*lambda*sin(lambda*x)+a^2*sin(lambda*x)^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\sqrt {2\,{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) +2} \left ( 2\,\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunC \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) a+\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunCPrime \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \lambda +2\,y\HeunC \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \right ) \left ( 2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) a+\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \HeunCPrime \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \lambda +2\,\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) a+\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunCPrime \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \lambda +2\,\sin \left ( \lambda \,x \right ) y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) +\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \lambda +2\,y\HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( y^2 + a \sin (\beta x) y + a b \sin (\beta x)-b^2 \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (y^2 + a*Sin[beta*x]*y + a*b*Sin[beta*x] - b^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+( y^2 + a*sin(beta*x)* y + a*b*sin(beta*x)-b^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b+y} \left ( b\int \!{{\rm e}^{-{\frac {2\,bx\beta +\cos \left ( \beta \,x \right ) a}{\beta }}}}\,{\rm d}x+y\int \!{{\rm e}^{-{\frac {2\,bx\beta +\cos \left ( \beta \,x \right ) a}{\beta }}}}\,{\rm d}x+{{\rm e}^{-{\frac {2\,bx\beta +\cos \left ( \beta \,x \right ) a}{\beta }}}} \right ) } \right ) \] contains unresolved integrals
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( y^2 + a x \sin ^m(b x) y + a \sin ^m(b x)\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (y^2 + a*x*Sin[b*x]^m*y + a*Sin[b*x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+( y^2 + a*x*sin(b*x)^m*y + a*sin(b*x)^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac {a \left ( \sin \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac {a \left ( \sin \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac {a \left ( \sin \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \] contains unresolved integrals
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left (\lambda \sin (\lambda x) y^2 + \lambda \sin ^3(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + lambda*Sin[lambda*x]^3)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(lambda*sin(lambda*x)*y^2 + lambda*sin(lambda*x)^3)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\sqrt {\pi } \left ( \cos \left ( \lambda \,x \right ) +y \right ) }{\sqrt {\pi }\cos \left ( \lambda \,x \right ) \erfi \left ( \cos \left ( \lambda \,x \right ) \right ) +\sqrt {\pi }\erfi \left ( \cos \left ( \lambda \,x \right ) \right ) y-2\,{{\rm e}^{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}}}}} \right ) \]
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ 2 w_x +\left ((\lambda +a-a \sin (\lambda x)) y^2 + \lambda -a -a \sin (\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = 2*D[w[x, y], x] + ((lambda + a - a*Sin[lambda*x])*y^2 + lambda - a - a*Sin[lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := 2*diff(w(x,y),x)+((lambda+a-a*sin(lambda*x))*y^2 +lambda -a -a*sin(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( { \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}-\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{5} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}-3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{2}\lambda +3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda +5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}ya{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}-4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-7\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{2}\lambda -5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}ya{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}+3\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{2}\lambda +3\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}ya{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}-\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda +4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}y{a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}ya{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{3}+\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\lambda }^{3}+7\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{2}\lambda +5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}ya{\lambda }^{2}-3\,\sin \left ( \lambda \,x \right ) y{a}^{2}\lambda -3\,\sin \left ( \lambda \,x \right ) ya{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{5}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{3}+3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}y{a}^{2}\lambda -3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}-3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}y{\lambda }^{3}-2\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}+ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda +2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}-\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda -2\,\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}y{a}^{3}-3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}y{a}^{3}+3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}y{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}y{\lambda }^{3}-\sin \left ( \lambda \,x \right ) y{a}^{3}-\sin \left ( \lambda \,x \right ) y{\lambda }^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda +3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-2\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda -4\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}-3\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -2\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}+2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda +4\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+3\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda +2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda -3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2} \right ) \left ( -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}-2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}+\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda +2\,\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}-\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+2\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}+\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}- \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda -3\,\sin \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{2}\lambda -3\,\sin \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}ya{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{3}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{3}+\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\lambda }^{3}+4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}ya{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda +\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}+2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}ya{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}+3\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{2}\lambda +3\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}ya{\lambda }^{2}+\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}+3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{3}-\sin \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{a}^{3}-\sin \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\lambda }^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda -3\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda -5\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}-\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda +\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}-2\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}+3\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda +5\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}-\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{a}^{2}\lambda -\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{\lambda }^{4}+2\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{\lambda }^{4}-2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{\lambda }^{4}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{\lambda }^{4}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {a}^{2}\lambda - \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sqrt {\sin \left ( \lambda \,x \right ) +1}y{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) a{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda -4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}+2\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda +6\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}+6\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}+\cos \left ( \lambda \,x \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}-6\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda -8\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}+6\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda +14\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}+10\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda +4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}-2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda -6\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}-6\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{5} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda -2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}-6\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda -4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}+6\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda +10\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}+4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}- \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}\int ^{\sin \left ( \lambda \,x \right ) }\!{\frac {{\it \_a}\,a-a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2}\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{\lambda }^{3}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{5}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda +2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}+6\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda +8\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}-6\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda -14\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}-10\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3} \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3}+6\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda +4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}-6\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{3}\lambda -10\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}{a}^{2}{\lambda }^{2}-4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}}a{\lambda }^{3} \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ((\lambda +a \sin ^2(\lambda x)) y^2 + \lambda -a +a \sin ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + ((lambda + a*Sin[lambda*x]^2)*y^2 + lambda - a + a*Sin[lambda*x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+((lambda+a*sin(lambda*x)^2)*y^2 + lambda -a +a*sin(lambda*x)^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1} \left ( 2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}ya-4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( 2\,\lambda \,x \right ) ya+\sin \left ( 2\,\lambda \,x \right ) \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}a+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}ya+2\, \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}y\lambda -2\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) a-2\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) \lambda -4\,\cos \left ( 2\,\lambda \,x \right ) y\lambda +a\sin \left ( 2\,\lambda \,x \right ) +2\,\lambda \,\sin \left ( 2\,\lambda \,x \right ) +2\,y\lambda \right ) \left ( -\sin \left ( 2\,\lambda \,x \right ) \sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}a-2\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}ya-2\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}y\lambda -4\,\sin \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }a\lambda +2\,\sin \left ( 2\,\lambda \,x \right ) \sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\cos \left ( 2\,\lambda \,x \right ) a+2\,\sin \left ( 2\,\lambda \,x \right ) \sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\cos \left ( 2\,\lambda \,x \right ) \lambda +4\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\cos \left ( 2\,\lambda \,x \right ) y\lambda -8\,\sin \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }{\lambda }^{2}-\sin \left ( 2\,\lambda \,x \right ) \sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xa-2\,\sin \left ( 2\,\lambda \,x \right ) \sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\lambda -2\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}xy\lambda -2\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}ya+4\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\int \!-2\,{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a-a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( 2\,\lambda \,x \right ) ya+4\,\sin \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }\cos \left ( 2\,\lambda \,x \right ) a\lambda \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x -\left ( (k+1) x^k y^2 - a x^{k+1}(\sin x)^m y + a (\sin x)^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] - ((k + 1)*x^k*y^2 - a*x^(k + 1)*Sin[x]^m*y + a*Sin[x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)-((k+1)*x^k*y^2 - a*x^(k+1)*(sin(x))^m*y + a*(sin(x))^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text {server hangs} \] (server hangs)
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Added January 14, 2019.
Problem 2.6.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( a \sin ^k(\lambda x + \mu )(y-b x^n -c)^2 + y - b x^n + b n x^{n-1} - c \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (a*Sin[lambda*x + mu]^k*(y - b*x^n - c)^2 + y - b*x^n + b*n*x^(n - 1) - c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(a*sin(lambda*x + mu)^k * (y-b*x^n -c)^2 + y - b*x^n + b*n*x^(n-1) - c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { Exception } \] Timed out
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Added January 14, 2019.
Problem 2.6.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x +\left ( a \sin ^m(\lambda x ) y^2 + k y + a b^2 x^{2 k} \sin ^m(\lambda x) \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = x*D[w[x, y], x] + (a*Sin[lambda*x]^m*y^2 + k*y + a*b^2*x^(2*k)*Sin[lambda*x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x a K[1]^{k-1} \sin ^m(\lambda K[1]) \, dK[1]\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := x*diff(w(x,y),x)+(a*sin(lambda*x)^m*y^2 + k*y + a*b^2*x^(2*k)*sin(lambda*x)^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( ab\int \! \left ( \sin \left ( \lambda \,x \right ) \right ) ^{m}{x}^{k-1}\,{\rm d}x-\arctan \left ( {\frac {{x}^{-k}y}{b}} \right ) \right ) \]
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Added January 14, 2019.
Problem 2.6.1.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a \sin (\lambda x) + b) w_x +\left ( y^2+ c \sin (\mu x) y - k^2 + c k \sin (\mu x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = (a*Sin[lambda*x] + b)*D[w[x, y], x] + (y^2 + c*Sin[mu*x]*y - k^2 + c*k*Sin[mu*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := (a *sin(lambda*x) + b)*diff(w(x,y),x)+(y^2+ c*sin(mu*x)* y - k^2 + c*k*sin(mu*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{k+y} \left ( k\int \!{\frac {1}{\sin \left ( \lambda \,x \right ) a+b}{{\rm e}^{{\frac {1}{\lambda \,\sqrt {-{a}^{2}+{b}^{2}}} \left ( c\int \!{\frac {\sin \left ( \mu \,x \right ) }{\sin \left ( \lambda \,x \right ) a+b}}\,{\rm d}x\lambda \,\sqrt {-{a}^{2}+{b}^{2}}-4\,k\arctan \left ( {\frac {b\sin \left ( 1/2\,\lambda \,x \right ) +a\cos \left ( 1/2\,\lambda \,x \right ) }{\sqrt {-{a}^{2}+{b}^{2}}\cos \left ( 1/2\,\lambda \,x \right ) }} \right ) \right ) }}}}\,{\rm d}x+y\int \!{\frac {1}{\sin \left ( \lambda \,x \right ) a+b}{{\rm e}^{{\frac {1}{\lambda \,\sqrt {-{a}^{2}+{b}^{2}}} \left ( c\int \!{\frac {\sin \left ( \mu \,x \right ) }{\sin \left ( \lambda \,x \right ) a+b}}\,{\rm d}x\lambda \,\sqrt {-{a}^{2}+{b}^{2}}-4\,k\arctan \left ( {\frac {b\sin \left ( 1/2\,\lambda \,x \right ) +a\cos \left ( 1/2\,\lambda \,x \right ) }{\sqrt {-{a}^{2}+{b}^{2}}\cos \left ( 1/2\,\lambda \,x \right ) }} \right ) \right ) }}}}\,{\rm d}x+{{\rm e}^{{\frac {1}{\lambda \,\sqrt {-{a}^{2}+{b}^{2}}} \left ( c\int \!{\frac {\sin \left ( \mu \,x \right ) }{\sin \left ( \lambda \,x \right ) a+b}}\,{\rm d}x\lambda \,\sqrt {-{a}^{2}+{b}^{2}}-4\,k\arctan \left ( {\frac {b\sin \left ( 1/2\,\lambda \,x \right ) +a\cos \left ( 1/2\,\lambda \,x \right ) }{\sqrt {-{a}^{2}+{b}^{2}}\cos \left ( 1/2\,\lambda \,x \right ) }} \right ) \right ) }}} \right ) } \right ) \]