____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \arctan ^k(\lambda x) + c_2 \arctan ^n(\beta y) \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcTan[lambda*x]^k+c2*ArcTan[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \text {\$Aborted} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*arctan(lambda*x)^k+c2*arctan(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c2}\, \left ( \arctan \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}+{\it c1}\, \left ( \arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}} \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \arctan ^k(\lambda x) \arctan ^n(\beta y) \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcTan[lambda*x]^k*ArcTan[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x \frac {e^{-\frac {c K[1]}{a}} \tan ^{-1}(\lambda K[1])^k \tan ^{-1}\left (\beta \left (\frac {b (K[1]-x)}{a}+y\right )\right )^n}{a} \, dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ arctan(lambda*x)^k*arctan(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}}{a} \left ( \arctan \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}} \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c_1 \arctan (\lambda _1 x) + c_2 \arctan (\lambda _2 y)\right ) w+ s_1 \arctan ^n(\beta _1 x)+ s_2 \arctan ^k(\beta _2 y) \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcTan[lambda1*x] + c2*ArcTan[lambda2*y])*w[x,y]+ s1*ArcTan[beta1*x]^n+ s2*ArcTan[beta2*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \text {\$Aborted} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arctan(lambda1*x) + c2*arctan(lambda2*y))*w(x,y)+ s1*arctan(beta1*x)^n+ s2*arctan(beta2*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{{\frac {1}{ab} \left ( - \left ( \left ( {\it \_a}-x \right ) b+ya \right ) {\it c2}\,\arctan \left ( {\frac {\lambda 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) -{\it c1}\,{\it \_a}\,\arctan \left ( \lambda 1\,{\it \_a} \right ) b \right ) }}} \left ( {{\it \_a}}^{2}{\lambda 1}^{2}+1 \right ) ^{1/2\,{\frac {{\it c1}}{a\lambda 1}}} \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) ^{2}{\lambda 2}^{2}+{a}^{2}}{{a}^{2}}} \right ) ^{1/2\,{\frac {{\it c2}}{b\lambda 2}}} \left ( {\it s2}\, \left ( \arctan \left ( {\frac {\beta 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{k}+{\it s1}\, \left ( \arctan \left ( \beta 1\,{\it \_a} \right ) \right ) ^{n} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) \left ( {\lambda 1}^{2}{x}^{2}+1 \right ) ^{-1/2\,{\frac {{\it c1}}{a\lambda 1}}} \left ( {\lambda 2}^{2}{y}^{2}+1 \right ) ^{-1/2\,{\frac {{\it c2}}{b\lambda 2}}}{{\rm e}^{{\frac {a\arctan \left ( \lambda 2\,y \right ) y{\it c2}+{\it c1}\,x\arctan \left ( \lambda 1\,x \right ) b}{ab}}}} \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arctan ^m(\mu x) w_y = c \arctan ^k(\nu x) w + p \arctan ^n(\beta y) \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*D[w[x, y], x] + b*ArcTan[mu*x]^m*D[w[x, y], y] == c*ArcTan[nu*x]^k*w[x,y]+p*ArcTan[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \text {\$Aborted} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*diff(w(x,y),x)+ b*arctan(mu*x)^m*diff(w(x,y),y) = c*arctan(nu*x)^k*w(x,y)+p*arctan(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p}{a} \left ( \arctan \left ( {\frac {\beta }{a} \left ( b\int \! \left ( \arctan \left ( {\it \_f}\,\mu \right ) \right ) ^{m}\,{\rm d}{\it \_f}+ \left ( -\int \!{\frac {b \left ( \arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{n}{{\rm e}^{-{\frac {c\int \! \left ( \arctan \left ( \nu \,{\it \_f} \right ) \right ) ^{k}\,{\rm d}{\it \_f}}{a}}}}}{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \arctan \left ( \nu \,x \right ) \right ) ^{k}c}{a}}\,{\rm d}x}} \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arctan ^m(\mu x) w_y = c \arctan ^k(\nu y) w + p \arctan ^n(\beta x) \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*D[w[x, y], x] + b*ArcTan[mu*x]^m*D[w[x, y], y] == c*ArcTan[nu*y]^k*w[x,y]+p*ArcTan[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \text {\$Aborted} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*diff(w(x,y),x)+ b*arctan(mu*x)^m*diff(w(x,y),y) = c*arctan(nu*y)^k*w(x,y)+p*arctan(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p \left ( \arctan \left ( \beta \,{\it \_f} \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \arctan \left ( {\frac {\nu }{a} \left ( b\int \! \left ( \arctan \left ( {\it \_f}\,\mu \right ) \right ) ^{m}\,{\rm d}{\it \_f}+ \left ( -\int \!{\frac {b \left ( \arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{k}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a} \left ( \arctan \left ( \nu \, \left ( \int \!{\frac {b \left ( \arctan \left ( {\it \_b}\,\mu \right ) \right ) ^{m}}{a}}\,{\rm d}{\it \_b}-\int \!{\frac {b \left ( \arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) \right ) \right ) ^{k}}{d{\it \_b}}}} \]