Added Oct 18, 2019.
Problem Chapter 8.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \cos ^n(\beta x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] + b*D[w[x,y,z],z]== c*Cos[beta*x]^n*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (-\frac {c \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)= c*cos(beta*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \left (\cos ^{n}\left (\beta x \right )\right )d x}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \cos (\beta z) w_z = \left ( k \cos (\lambda x)+s \cos (\gamma y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Cos[beta*z]*D[w[x,y,z],z]== (k*Cos[lambda*x]+s*Cos[gamma*y])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {k \sin (\lambda x)}{a \lambda }+\frac {s \sin (\gamma y)}{b \gamma }} c_1\left (y-\frac {b x}{a},-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta z) \left (2 \left (2 \sec (\beta z) \sqrt {\sin ^2(\beta z) \cos ^2(\beta z) \sinh ^2\left (\frac {\beta c x}{a}\right ) \left (\cosh \left (\frac {4 \beta c x}{a}\right )-\sinh \left (\frac {4 \beta c x}{a}\right )\right )}+\sinh ^3\left (\frac {\beta c x}{a}\right )+\sinh \left (\frac {\beta c x}{a}\right )\right )-2 \cosh ^3\left (\frac {\beta c x}{a}\right )+\left (1-3 \cosh \left (\frac {2 \beta c x}{a}\right )\right ) \cosh \left (\frac {\beta c x}{a}\right )+6 \sinh \left (\frac {\beta c x}{a}\right ) \cosh ^2\left (\frac {\beta c x}{a}\right )\right )}{4 \cosh \left (\frac {2 \beta c x}{a}\right )-4 \sinh \left (\frac {2 \beta c x}{a}\right )}\right )}{\beta }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+ c*cos(beta*z)*diff(w(x,y,z),z)= (k*cos(lambda*x)+s*cos(gamma*y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {a \ln \left (\RootOf \left (\beta z -\arctan \left (\frac {\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 \beta c x}{a}}-1}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 \beta c x}{a}}+1}, \frac {2 \mathit {\_Z} \,{\mathrm e}^{\frac {\beta c x}{a}}}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 \beta c x}{a}}+1}\right )\right )\right )}{\beta c}\right ) {\mathrm e}^{\frac {a \lambda s \sin \left (\gamma y \right )+b \gamma k \sin \left (\lambda x \right )}{a b \gamma \lambda }}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \cos ^n(\beta x) w_y + b \cos ^k(\lambda x) w_z = c \cos ^m(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cos[beta*x]^n*D[w[x, y,z], y] + b*Cos[lambda*x]^k*D[w[x,y,z],z]== c*Cos[gamma*x]^m*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (-\frac {c \sqrt {\sin ^2(\gamma x)} \csc (\gamma x) \cos ^{m+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (\frac {a \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\beta x)\right )}{\beta n+\beta }+y,\frac {b \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cos ^2(\lambda x)\right )}{k \lambda +\lambda }+z\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*cos(beta*x)^n*diff(w(x,y,z),y)+ b*cos(lambda*x)^k*diff(w(x,y,z),z)= c*cos(gamma*x)^m*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int a \left (\cos ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int b \left (\cos ^{k}\left (\lambda x \right )\right )d x \right )\right ) {\mathrm e}^{\int c \left (\cos ^{m}\left (\gamma x \right )\right )d x}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \cos ^n(\beta x) w_y + b \cos ^m(\gamma y) w_z = \left ( c \cos ^k(\gamma y) + s \cos ^r(\mu z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cos[beta*x]^n*D[w[x, y,z], y] + b*Cos[gamma*y]^m*D[w[x,y,z],z]== (c*Cos[gamma*y]^k+s*Cos[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)+a*cos(beta*x)^n*diff(w(x,y,z),y)+ b*cos(gamma*y)^m*diff(w(x,y,z),z)= (c*cos(gamma*y)^k+s*cos(mu*z)^r)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int a \left (\cos ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int _{}^{x}b \left (\cos ^{m}\left (\left (a \left (\int \left (\cos ^{n}\left (\mathit {\_b} \beta \right )\right )d \mathit {\_b} \right )+y -\left (\int a \left (\cos ^{n}\left (\beta x \right )\right )d x \right )\right ) \gamma \right )\right )d\mathit {\_b} \right )\right ) {\mathrm e}^{\int _{}^{x}\left (c \left (\cos ^{k}\left (\left (-y -\left (\int a \left (\cos ^{n}\left (\mathit {\_g} \beta \right )\right )d \mathit {\_g} \right )+\int a \left (\cos ^{n}\left (\beta x \right )\right )d x \right ) \gamma \right )\right )+s \left (\cos ^{r}\left (\left (z +\int b \left (\cos ^{m}\left (\left (a \left (\int \left (\cos ^{n}\left (\mathit {\_g} \beta \right )\right )d \mathit {\_g} \right )+y -\left (\int a \left (\cos ^{n}\left (\beta x \right )\right )d x \right )\right ) \gamma \right )\right )d \mathit {\_g} -\left (\int ^{x}b \left (\cos ^{m}\left (\left (a \left (\int \left (\cos ^{n}\left (\mathit {\_b} \beta \right )\right )d \mathit {\_b} \right )+y -\left (\int a \left (\cos ^{n}\left (\beta x \right )\right )d x \right )\right ) \gamma \right )\right )d \mathit {\_b} \right )\right ) \mu \right )\right )\right )d\mathit {\_g}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \cos (\beta y) w_y + c \cos (\lambda x) w_z = k \cos (\gamma z) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cos[beta*y]*D[w[x, y,z], y] + c*Cos[lambda*x]^m*D[w[x,y,z],z]== k*Cos[gamma*z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\begin {align*} & \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right )}{a \lambda m+a \lambda }+z\right ) \exp \left (\int _1^x\frac {k \cos \left (\frac {\gamma \left (c \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cos ^{m+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{a \lambda (m+1)}\right )}{a}dK[1]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right )}{a \lambda m+a \lambda }+z\right ) \exp \left (\int _1^x\frac {k \cos \left (\frac {\gamma \left (c \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cos ^{m+1}(\lambda K[2]) \csc (\lambda K[2]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda K[2])\right ) \sqrt {\sin ^2(\lambda K[2])}\right )}{a \lambda (m+1)}\right )}{a}dK[2]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right )}{a \lambda m+a \lambda }+z\right ) \exp \left (\int _1^x\frac {k \cos \left (\frac {\gamma \left (c \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cos ^{m+1}(\lambda K[3]) \csc (\lambda K[3]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda K[3])\right ) \sqrt {\sin ^2(\lambda K[3])}\right )}{a \lambda (m+1)}\right )}{a}dK[3]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right )}{a \lambda m+a \lambda }+z\right ) \exp \left (\int _1^x\frac {k \cos \left (\frac {\gamma \left (c \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cos ^{m+1}(\lambda K[4]) \csc (\lambda K[4]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda K[4])\right ) \sqrt {\sin ^2(\lambda K[4])}\right )}{a \lambda (m+1)}\right )}{a}dK[4]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right )}{a \lambda m+a \lambda }+z\right ) \exp \left (\int _1^x\frac {k \cos \left (\frac {\gamma \left (c \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cos ^{m+1}(\lambda K[5]) \csc (\lambda K[5]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda K[5])\right ) \sqrt {\sin ^2(\lambda K[5])}\right )}{a \lambda (m+1)}\right )}{a}dK[5]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right )}{a \lambda m+a \lambda }+z\right ) \exp \left (\int _1^x\frac {k \cos \left (\frac {\gamma \left (c \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cos ^{m+1}(\lambda K[6]) \csc (\lambda K[6]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda K[6])\right ) \sqrt {\sin ^2(\lambda K[6])}\right )}{a \lambda (m+1)}\right )}{a}dK[6]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right )}{a \lambda m+a \lambda }+z\right ) \exp \left (\int _1^x\frac {k \cos \left (\frac {\gamma \left (c \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cos ^{m+1}(\lambda K[7]) \csc (\lambda K[7]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda K[7])\right ) \sqrt {\sin ^2(\lambda K[7])}\right )}{a \lambda (m+1)}\right )}{a}dK[7]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right )}{a \lambda m+a \lambda }+z\right ) \exp \left (\int _1^x\frac {k \cos \left (\frac {\gamma \left (c \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cos ^{m+1}(\lambda K[8]) \csc (\lambda K[8]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\lambda K[8])\right ) \sqrt {\sin ^2(\lambda K[8])}\right )}{a \lambda (m+1)}\right )}{a}dK[8]\right )\right \}\\ \end {align*}
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*cos(beta*y)*diff(w(x,y,z),y)+ c*cos(lambda*x)^m*diff(w(x,y,z),z)= k*cos(gamma*z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a \ln \left (\RootOf \left (\beta y -\arctan \left (\frac {\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}-1}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}, \frac {2 \mathit {\_Z} \,{\mathrm e}^{\frac {b \beta x}{a}}}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}\right )\right )\right )}{b \beta }, z -\left (\int \frac {c \left (\cos ^{m}\left (\lambda x \right )\right )}{a}d x \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {k \cos \left (\left (-z -\left (\int \frac {c \left (\cos ^{m}\left (\mathit {\_b} \lambda \right )\right )}{a}d \mathit {\_b} \right )+\int \frac {c \left (\cos ^{m}\left (\lambda x \right )\right )}{a}d x \right ) \gamma \right )}{a}d\mathit {\_b}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \cos ^{n_1}(\lambda _1 x) w_x + b_1 \cos ^{m_1}(\beta _1 y) w_y + c_1 \cos ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \cos ^{n_2}(\lambda _2 x) + b_2 \cos ^{m_2}(\beta _2 y) + c_2 \cos ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Cos[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cos[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*Cos[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*cos(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*cos(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*cos(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 (\cos ^{-\mathit {m1}}\left (\beta 1 y \right )\right )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 (\cos ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right ) \left (\cos ^{\mathit {n1}}\left (\lambda 1 \RootOf \left (\int \left (\cos ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cos ^{-\mathit {m1}}\left (\beta 1 y \right )\right )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 (\cos ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cos ^{-\mathit {m1}}\left (\beta 1 y \right )\right )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 {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )\right )+\mathit {b2} \left (\cos ^{\mathit {m2}}\left (\mathit {\_f} \beta 2 \right )\right )+\mathit {c2} \left (\cos ^{\mathit {k2}}\left (\gamma 2 \RootOf \left (\int \left (\cos ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cos ^{-\mathit {m1}}\left (\beta 1 y \right )\right )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 {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )\right )\right ) \left (\cos ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )}{\mathit {b1}}d\mathit {\_f}}\]
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