Added May 26, 2019.
Problem Chapter 6.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \sin (\gamma z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*Sin[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log \left (\tan \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*sin(gamma*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {a}{c\gamma }\ln \left ( \RootOf \left ( \gamma \,z-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {c\gamma \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}},-{ \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sin (\beta y) w_y + c \sin (\lambda x) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Sin[beta*y]*D[w[x, y,z], y] +c*Sin[lambda*x]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {c \cos (\lambda x)}{a \lambda }+z,\frac {\log \left (\tan \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*sin(beta*y)*diff(w(x,y,z),y)+c*sin(lambda*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a}{\beta \,b}\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {b\beta \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}},-{ \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) },{\frac {za\lambda +c\cos \left ( x\lambda \right ) }{a\lambda }} \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sin (\beta y) w_y + c \sin (\gamma z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Sin[beta*y]*D[w[x, y,z], y] +c*Sin[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tan \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},\frac {\log \left (\tan \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*sin(beta*y)*diff(w(x,y,z),y)+c*sin(gamma*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a}{\beta \,b}\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {b\beta \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}},-{ \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) },{\frac {a}{c\gamma }\ln \left ( \RootOf \left ( \gamma \,z-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {c\gamma \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}},-{ \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sin (\lambda x) \sin (\beta y) w_y + c w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Sin[lambda*x]*Sin[beta*y]*D[w[x, y,z], y] +c*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (z-\frac {c x}{a},\frac {a \log \left (\tan ^2\left (\frac {\beta y}{2}\right )\right )}{\beta }+\frac {2 b \cos (\lambda x)}{\lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*sin(lambda*x)*sin(beta*y)*diff(w(x,y,z),y)+c*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{b\beta \,\lambda } \left ( -\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( 2\,{\frac {{\it \_Z}}{{{\it \_Z}}^{2}+1}},{\frac {{{\it \_Z}}^{2}-1}{{{\it \_Z}}^{2}+1}} \right ) \right ) \right ) a\lambda +\cos \left ( x\lambda \right ) b\beta \right ) },{\frac {za-cx}{a}} \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sin ^n(\lambda x) \sin ^m(\beta y) w_y + c \sin ^k(\mu x) \sin ^r(\gamma *z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Sin[lambda*x]^n*Sin[beta*y]^m*D[w[x, y,z], y] +c*Sin[mu*x]^k*Sin[gamma*z]^r*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\sqrt {\cos ^2(\beta y)} \sec (\beta y) \sin ^{1-m}(\beta y) \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\sin ^2(\beta y)\right )}{\beta -\beta m}-\frac {b \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{a \lambda n+a \lambda },\frac {\sqrt {\cos ^2(\gamma z)} \sec (\gamma z) \sin ^{1-r}(\gamma z) \, _2F_1\left (\frac {1}{2},\frac {1-r}{2};\frac {3-r}{2};\sin ^2(\gamma z)\right )}{\gamma -\gamma r}-\frac {c \sqrt {\cos ^2(\mu x)} \sec (\mu x) \sin ^{k+1}(\mu x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\mu x)\right )}{a k \mu +a \mu }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*sin(lambda*x)^n*sin(beta*y)^m*diff(w(x,y,z),y)+c*sin(mu*x)^k*sin(gamma*z)^r*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+\int \!{\frac { \left ( \sin \left ( \beta \,y \right ) \right ) ^{-m}a}{b}}\,{\rm d}y,-\int \! \left ( \sin \left ( \mu \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \sin \left ( \gamma \,z \right ) \right ) ^{-r}a}{c}}\,{\rm d}z \right ) \]
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