____________________________________________________________________________________
Added January 10, 2019.
Problem 2.4.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \sinh (\lambda x) \cosh (\mu y) w_y = 0 \]
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*Sinh[lambda*x]*Cosh[mu*y]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {2 \left (a \mu \cosh (\lambda x)-2 \lambda \tan ^{-1}\left (\tanh \left (\frac {\mu y}{2}\right )\right )\right )}{\lambda \mu }\right )\right \}\right \} \]
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';s:='s';B:='B';mu:='mu';d:='d'; pde := diff(w(x,y),x)+a*sinh(lambda*x)*cosh(mu*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\cosh \left ( \lambda \,x \right ) \mu \,a-2\,\arctan \left ( {{\rm e}^{\mu \,y}} \right ) \lambda }{\lambda \,\mu \,a}} \right ) \]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.4.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \cosh (\lambda x) \sinh (\mu y) w_y = 0 \]
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*Cosh[lambda*x]*Sinh[mu*y]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\lambda \log \left (\tanh ^2\left (\frac {\mu y}{2}\right )\right )-2 a \mu \sinh (\lambda x)}{\lambda \mu }\right )\right \}\right \} \]
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';s:='s';B:='B';mu:='mu';d:='d'; pde := diff(w(x,y),x)+a*cosh(lambda*x)*sinh(mu*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {{{\rm e}^{\lambda \,x}}a\mu -\mu \,a{{\rm e}^{-\lambda \,x}}+4\,\arctanh \left ( {{\rm e}^{\mu \,y}} \right ) \lambda }{\lambda \,\mu \,a}} \right ) \]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.4.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2 -2 \lambda ^2 \tanh ^2(\lambda x) - 2 \lambda ^2 \coth ^2(\lambda x) \right ) w_y = 0 \]
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 - 2*lambda^2*Tanh[lambda*x]^2 - 2*lambda^2*Coth[lambda*x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-4 \lambda x} \left (16 \lambda ^2 x e^{4 \lambda x}+16 \lambda ^2 x e^{8 \lambda x}+8 \lambda x y e^{4 \lambda x}-8 \lambda x y e^{8 \lambda x}-y e^{4 \lambda x}-y e^{8 \lambda x}+y e^{12 \lambda x}+14 \lambda e^{4 \lambda x}-14 \lambda e^{8 \lambda x}+2 \lambda e^{12 \lambda x}-2 \lambda +y\right )}{2 \left (-y e^{4 \lambda x}+2 \lambda e^{4 \lambda x}+2 \lambda +y\right )}\right )\right \}\right \} \]
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';s:='s';B:='B';mu:='mu';d:='d'; pde := diff(w(x,y),x)+(y^2 -2 *lambda^2*tanh(lambda*x)^2 - 2*lambda^2*coth(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 ( -4\,{(\sinh \left ( \lambda \,x \right ) \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\lambda +\sinh \left ( \lambda \,x \right ) {\rm coth} \left (\lambda \,x\right )y-2\,\cosh \left ( \lambda \,x \right ) {\rm coth} \left (\lambda \,x\right )\lambda -\sinh \left ( \lambda \,x \right ) \lambda ) \left ( 4\,\sinh \left ( \lambda \,x \right ) \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\ln \left ( -{\frac {-\cosh \left ( \lambda \,x \right ) +\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda -4\,\sinh \left ( \lambda \,x \right ) \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\ln \left ( {\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda +4\,\sinh \left ( \lambda \,x \right ) {\rm coth} \left (\lambda \,x\right )\ln \left ( -{\frac {-\cosh \left ( \lambda \,x \right ) +\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) y-4\,\sinh \left ( \lambda \,x \right ) {\rm coth} \left (\lambda \,x\right )\ln \left ( {\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) y-8\,\cosh \left ( \lambda \,x \right ) {\rm coth} \left (\lambda \,x\right )\ln \left ( -{\frac {-\cosh \left ( \lambda \,x \right ) +\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda +8\,\cosh \left ( \lambda \,x \right ) {\rm coth} \left (\lambda \,x\right )\ln \left ( {\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda - \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\cosh \left ( 3\,\lambda \,x \right ) \lambda + \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{2}\cosh \left ( 5\,\lambda \,x \right ) \lambda -8\,\sinh \left ( \lambda \,x \right ) {\rm coth} \left (\lambda \,x\right )\lambda -4\,\sinh \left ( \lambda \,x \right ) \ln \left ( -{\frac {-\cosh \left ( \lambda \,x \right ) +\sinh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda +4\,\sinh \left ( \lambda \,x \right ) \ln \left ( {\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) \lambda -{\rm coth} \left (\lambda \,x\right )\cosh \left ( 3\,\lambda \,x \right ) y+{\rm coth} \left (\lambda \,x\right )y\cosh \left ( 5\,\lambda \,x \right ) -6\,{\rm coth} \left (\lambda \,x\right )\sinh \left ( 3\,\lambda \,x \right ) \lambda +2\,{\rm coth} \left (\lambda \,x\right )\sinh \left ( 5\,\lambda \,x \right ) \lambda +\cosh \left ( 3\,\lambda \,x \right ) \lambda -\cosh \left ( 5\,\lambda \,x \right ) \lambda \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2 +\lambda (a+b)-2 a b -a(a+\lambda ) \tanh ^2(\lambda x) - b(b+\lambda ) \coth ^2(\lambda x) \right ) w_y = 0 \]
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 + lambda*(a + b) - 2*a*b - a*(a + lambda)*Tanh[lambda*x]^2 - b*(b + lambda)*Coth[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';s:='s';B:='B';mu:='mu';d:='d'; pde := diff(w(x,y),x)+(y^2 +lambda*(a+b)-2*a*b -a*(a+lambda)*tanh(lambda*x)^2 - b*(b+lambda)*coth(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 ( { \left ( -2\,a-3\,\lambda \right ) \left ( \sinh \left ( \lambda \,x \right ) \cosh \left ( \lambda \,x \right ) y-a \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}- \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}b+a \right ) \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2} \left ( {\frac {\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) ^{-{\frac {2\,a+\lambda }{\lambda }}} \left ( - \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{-2} \right ) ^{{\frac {a+b}{\lambda }}} \left ( 2\,\sinh \left ( \lambda \,x \right ) \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{3}y{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a+3\,\sinh \left ( \lambda \,x \right ) \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{3}y{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}\lambda +2\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{a}^{2}+2\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}ab+3\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a\lambda +3\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{4}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}b\lambda -2\,\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a\sinh \left ( \lambda \,x \right ) -3\,\cosh \left ( \lambda \,x \right ) y{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}\lambda \,\sinh \left ( \lambda \,x \right ) -4\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{a}^{2}-2\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}ab-8\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a\lambda -3\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}b\lambda -3\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{\lambda }^{2}+4\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(2,-1/2\,{\frac {2\,b-3\,\lambda }{\lambda }};\,1/2\,{\frac {2\,a+5\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}b\lambda -2\, \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(2,-1/2\,{\frac {2\,b-3\,\lambda }{\lambda }};\,1/2\,{\frac {2\,a+5\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{\lambda }^{2}+2\,{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{a}^{2}+5\,{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}a\lambda +3\,{\mbox {$_2$F$_1$}(1,-1/2\,{\frac {2\,b-\lambda }{\lambda }};\,1/2\,{\frac {2\,a+3\,\lambda }{\lambda }};\,{\frac { \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{2}}{ \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}}})}{\lambda }^{2} \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.4.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \sinh (\lambda y) w_x + a \cosh (\beta x) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = Sinh[lambda*y]*D[w[x, y], x] + a*Cosh[beta*x]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\beta \cosh (\lambda y)-a \lambda \sinh (\beta x)}{\beta \lambda }\right )\right \}\right \} \]
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';s:='s';B:='B';mu:='mu';d:='d'; pde := sinh(lambda*y)*diff(w(x,y),x)+a*cosh(beta*x)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-\sinh \left ( \beta \,x \right ) a\lambda +\cosh \left ( y\lambda \right ) \beta }{a\beta \,\lambda }} \right ) \]
____________________________________________________________________________________
Added January 10, 2019.
Problem 2.4.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \left ( a x^n \cosh ^m(\lambda y)+ b x \right ) w_x + \sinh ^k(\beta y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = (a*x^n*Cosh[lambda*y]^m + b*x)*D[w[x, y], x] + Sinh[beta*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';s:='s';B:='B';mu:='mu';d:='d'; pde := (a*x^n*cosh(lambda*y)^m+b*x)*diff(w(x,y),x)+sinh(beta*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \sinh \left ( \beta \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \sinh \left ( \beta \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cosh \left ( y\lambda \right ) \right ) ^{m} \left ( \sinh \left ( \beta \,y \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \sinh \left ( \beta \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cosh \left ( y\lambda \right ) \right ) ^{m} \left ( \sinh \left ( \beta \,y \right ) \right ) ^{-k}\,{\rm d}y \right ) \]