Added Oct 18, 2019.
Problem Chapter 8.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \sin ^n(\lambda x) w_y + b \cos ^m(\beta x) w_z = c \sin ^k(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Cos[beta*x]^m*D[w[x,y,z],z]== c*Sin[gamma*x]^k*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \sqrt {\cos ^2(\gamma x)} \sec (\gamma x) \sin ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\gamma x)\right )}{\gamma k+\gamma }\right ) c_1\left (\frac {b \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\beta x)\right )}{\beta m+\beta }+z,y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*cos(beta*x)^m*diff(w(x,y,z),z)= c*sin(gamma*x)^k*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int b \left (\cos ^{m}\left (\beta x \right )\right )d x \right )\right ) {\mathrm e}^{\int c \left (\sin ^{k}\left (\gamma x \right )\right )d x}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \cos ^n(\lambda x)w_y + b \sin ^m(\beta y) w_z = \left ( c \cos ^k(\gamma y)+s \sin ^r(\mu z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cos[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*y]^m*D[w[x,y,z],z]== (c*Cos[gamma*y]^k+s*Sin[mu*z]^r)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+b*cos(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*y)^m*diff(w(x,y,z),z)= (c*cos(gamma*y)^k+s*sin(mu*z)^r)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int b \left (\cos ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int _{}^{x}b \left (\sin ^{m}\left (\left (b \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )+y -\left (\int b \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )\right )d\mathit {\_b} \right )\right ) {\mathrm e}^{\int _{}^{x}\left (c \left (\cos ^{k}\left (\left (-y -\left (\int b \left (\cos ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )+\int b \left (\cos ^{n}\left (\lambda x \right )\right )d x \right ) \gamma \right )\right )+s \left (\sin ^{r}\left (\left (z +\int b \left (\sin ^{m}\left (\left (b \left (\int \left (\cos ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )+y -\left (\int b \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )\right )d \mathit {\_g} -\left (\int ^{x}b \left (\sin ^{m}\left (\left (b \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )+y -\left (\int b \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )\right )d \mathit {\_b} \right )\right ) \mu \right )\right )\right )d\mathit {\_g}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \cos ^n(\lambda x) w_y + b \tan ^m(\beta y) w_z = \left ( c \cos ^k(\gamma y) + s \tan ^k(\mu z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cos[lambda*x]^n*D[w[x, y,z], y] + b*Tan[beta*y]^m*D[w[x,y,z],z]== (c*Cos[gamma*y]^k+s*Tan[mu*z]^k)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*cos(lambda*x)^n*diff(w(x,y,z),y)+ b*tan(beta*y)^m*diff(w(x,y,z),z)= (c*cos(gamma*y)^k+s*tan(mu*z)^k)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int _{}^{x}b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )-1}\right )^{m}d\mathit {\_b} \right )\right ) {\mathrm e}^{\int _{}^{x}\left (c \left (\cos ^{k}\left (\left (-y -\left (\int a \left (\cos ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )+\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right ) \gamma \right )\right )+s \left (\frac {\sin \left (\left (z +\int b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )\right )-1}\right )^{m}d \mathit {\_g} -\left (\int _{}^{x}b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )-1}\right )^{m}d\mathit {\_b} \right )\right ) \mu \right )}{\cos \left (\left (z +\int b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )\right )-1}\right )^{m}d \mathit {\_g} -\left (\int _{}^{x}b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )-1}\right )^{m}d\mathit {\_b} \right )\right ) \mu \right )}\right )^{k}\right )d\mathit {\_g}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \sin ^{n_1}(\lambda _1 x) w_x + b_1 \cot ^{m_1}(\beta _1 y) w_y + c_1 \cos ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \cos ^{n_2}(\lambda _2 x) + b_2 \sin ^{m_2}(\beta _2 y) + c_2 \cos ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Sin[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cot[beta1*y]^m1*D[w[x, y,z], y] + c1*Cos[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Cos[lambda2*z]^n2 + b2*Sin[beta2*y]^m2 + c2*Cos[gamma2*z]^k2)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*sin(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*cot(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cos(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*cos(lambda2*z)^n2 + b2*sin(beta2*y)^m2 + c2*cos(gamma2*z)^k2)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\left (\int \left (\frac {\cos \left (\beta 1 y \right )}{\sin \left (\beta 1 y \right )}\right )^{-\mathit {m1}}d y \right )+\int \frac {\mathit {b1} \left (\cos ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z , x -\left (\int _{}^{y}\frac {\mathit {a1} \left (\frac {\cos \left (\mathit {\_f} \beta 1 \right )}{\sin \left (\mathit {\_f} \beta 1 \right )}\right )^{-\mathit {m1}} \left (\sin ^{\mathit {n1}}\left (\lambda 1 \RootOf \left (\int \left (\frac {\cos \left (\mathit {\_f} \beta 1 \right )}{\sin \left (\mathit {\_f} \beta 1 \right )}\right )^{-\mathit {m1}}d \mathit {\_f} -\left (\int \left (\frac {\cos \left (\beta 1 y \right )}{\sin \left (\beta 1 y \right )}\right )^{-\mathit {m1}}d y \right )+\int \frac {\mathit {b1} \left (\cos ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cos ^{-\mathit {k1}}\left (\mathit {\_b} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_b} \right )\right )\right )\right )}{\mathit {b1}}d\mathit {\_f} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (\mathit {a2} \left (\cos ^{\mathit {n2}}\left (\lambda 2 \RootOf \left (\int \left (\frac {\cos \left (\beta 1 y \right )}{\sin \left (\beta 1 y \right )}\right )^{-\mathit {m1}}d y -\left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} \right )-\left (\int \frac {\mathit {b1} \left (\cos ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z \right )+\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cos ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )+\mathit {b2} \left (\sin ^{\mathit {m2}}\left (\mathit {\_f} \beta 2 \right )\right )+\mathit {c2} \left (\cos ^{\mathit {k2}}\left (\gamma 2 \RootOf \left (\int \left (\frac {\cos \left (\beta 1 y \right )}{\sin \left (\beta 1 y \right )}\right )^{-\mathit {m1}}d y -\left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} \right )-\left (\int \frac {\mathit {b1} \left (\cos ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z \right )+\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cos ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )\right ) \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )}{\mathit {b1}}d\mathit {\_f}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \tan ^{n_1}(\lambda _1 x) w_x + b_1 \cot ^{m_1}(\beta _1 y) w_y + c_1 \cot ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \cot ^{n_2}(\lambda _2 x) + b_2 \tan ^{m_2}(\beta _2 y) + c_2 \cot ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Tan[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cot[beta1*y]^m1*D[w[x, y,z], y] + c1*Cot[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Cot[lambda2*z]^n2 + b2*Tan[beta2*y]^m2 + c2*Cot[gamma2*z]^k2)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*tan(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*cot(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cot(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*cot(lambda2*z)^n2 + b2*tan(beta2*y)^m2 + c2*cot(gamma2*z)^k2)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z , x -\left (\int _{}^{y}\frac {\mathit {a1} \left (\frac {\cos \left (\mathit {\_f} \beta 1 \right )}{\sin \left (\mathit {\_f} \beta 1 \right )}\right )^{-\mathit {m1}} \left (\frac {\sin \left (\lambda 1 \RootOf \left (\int \left (\frac {\cos \left (\mathit {\_f} \beta 1 \right )}{\sin \left (\mathit {\_f} \beta 1 \right )}\right )^{-\mathit {m1}}d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {b1} \left (\frac {\cos \left (\mathit {\_b} \gamma 1 \right )}{\sin \left (\mathit {\_b} \gamma 1 \right )}\right )^{-\mathit {k1}}}{\mathit {c1}}d\mathit {\_b} \right )\right )\right )}{\cos \left (\lambda 1 \RootOf \left (\int \left (\frac {\cos \left (\mathit {\_f} \beta 1 \right )}{\sin \left (\mathit {\_f} \beta 1 \right )}\right )^{-\mathit {m1}}d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {b1} \left (\frac {\cos \left (\mathit {\_b} \gamma 1 \right )}{\sin \left (\mathit {\_b} \gamma 1 \right )}\right )^{-\mathit {k1}}}{\mathit {c1}}d\mathit {\_b} \right )\right )\right )}\right )^{\mathit {n1}}}{\mathit {b1}}d\mathit {\_f} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (\mathit {a2} \left (\frac {\cos \left (\lambda 2 \RootOf \left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d\mathit {\_a} \right )\right )\right )}{\sin \left (\lambda 2 \RootOf \left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d\mathit {\_a} \right )\right )\right )}\right )^{\mathit {n2}}+\mathit {b2} \left (\frac {\sin \left (\mathit {\_f} \beta 2 \right )}{\cos \left (\mathit {\_f} \beta 2 \right )}\right )^{\mathit {m2}}+\mathit {c2} \left (\frac {\cos \left (\gamma 2 \RootOf \left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d\mathit {\_a} \right )\right )\right )}{\sin \left (\gamma 2 \RootOf \left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d\mathit {\_a} \right )\right )\right )}\right )^{\mathit {k2}}\right ) \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )}{\mathit {b1}}d\mathit {\_f}}\]
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