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]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (-\frac {c \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{n+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( \cos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {a}{c\beta }\ln \left ( \RootOf \left ( \beta \,z-\arctan \left ( { \left ( {{\rm e}^{2\,{\frac {xc\beta }{a}}}}{{\it \_Z}}^{2}-1 \right ) \left ( {{\rm e}^{2\,{\frac {xc\beta }{a}}}}{{\it \_Z}}^{2}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {xc\beta }{a}}}} \left ( {{\rm e}^{2\,{\frac {xc\beta }{a}}}}{{\it \_Z}}^{2}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) {{\rm e}^{{\frac {k\sin \left ( \lambda \,x \right ) \gamma \,b+sa\lambda \,\sin \left ( \gamma \,y \right ) }{a\lambda \,\gamma \,b}}}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to \exp \left (-\frac {c \sqrt {\sin ^2(\gamma x)} \csc (\gamma x) \cos ^{m+1}(\gamma x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (\frac {a \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{n+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\beta x)\right )}{\beta n+\beta }+y,\frac {b \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(\lambda x)\right )}{k \lambda +\lambda }+z\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \cos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cos \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) {{\rm e}^{\int \!c \left ( \cos \left ( \gamma \,x \right ) \right ) ^{m}\,{\rm d}x}}\]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \cos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \cos \left ( \gamma \, \left ( \int \! \left ( \cos \left ( \beta \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b}a-\int \!a \left ( \cos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) {{\rm e}^{\int ^{x}\! \left ( \cos \left ( \mu \, \left ( \int \!b \left ( \cos \left ( \gamma \, \left ( a\int \! \left ( \cos \left ( \beta \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g}-\int \!a \left ( \cos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}\,{\rm d}{\it \_g}-\int ^{x}\!b \left ( \cos \left ( \gamma \, \left ( \int \! \left ( \cos \left ( \beta \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b}a-\int \!a \left ( \cos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{r}s+c \left ( \cos \left ( \gamma \, \left ( -\int \!a \left ( \cos \left ( \beta \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g}+\int \!a \left ( \cos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x-y \right ) \right ) \right ) ^{k}{d{\it \_g}}}}\]
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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) \, _2F_1\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) \, _2F_1\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]) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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]) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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]) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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]) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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]) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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]) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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]) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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]) \, _2F_1\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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a}{b\beta }\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( { \left ( {{\rm e}^{2\,{\frac {xb\beta }{a}}}}{{\it \_Z}}^{2}-1 \right ) \left ( {{\rm e}^{2\,{\frac {xb\beta }{a}}}}{{\it \_Z}}^{2}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {xb\beta }{a}}}} \left ( {{\rm e}^{2\,{\frac {xb\beta }{a}}}}{{\it \_Z}}^{2}+1 \right ) ^{-1}} \right ) \right ) \right ) },-\int \!{\frac {c \left ( \cos \left ( \lambda \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+z \right ) {{\rm e}^{\int ^{x}\!{\frac {k}{a}\cos \left ( \gamma \, \left ( -\int \!{\frac {c \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{m}}{a}}\,{\rm d}{\it \_b}+\int \!{\frac {c \left ( \cos \left ( \lambda \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-z \right ) \right ) }{d{\it \_b}}}}\]
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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 ) ={\it \_F1} \left ( -\int \! \left ( \cos \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z,-\int ^{y}\!{\frac {{\it a1}\, \left ( \cos \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( \cos \left ( \lambda 1\,\RootOf \left ( \int \! \left ( \cos \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \gamma 1\,{\it \_b} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_b}}-\int \! \left ( \cos \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n1}}}{d{\it \_f}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \cos \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( {\it a2}\, \left ( \cos \left ( \lambda 2\,\RootOf \left ( \int \! \left ( \cos \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cos \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n2}}+{\it c2}\, \left ( \cos \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \cos \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cos \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}}+{\it b2}\, \left ( \cos \left ( \beta 2\,{\it \_f} \right ) \right ) ^{{\it m2}} \right ) }{d{\it \_f}}}}\]
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