Added May 26, 2019.
Problem Chapter 6.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \tan (\gamma z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*Tan[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log (\sin (\gamma z))}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*tan(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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {a \ln \left (\frac {\tan \left (\gamma z \right )}{\sqrt {\tan ^{2}\left (\gamma z \right )+1}}\right )-\gamma c x}{\gamma c}\right )\]
Hand solution
Solve \begin {equation} aw_{x}+bw_{y}+c\tan \left ( \gamma z\right ) w_{z}=0 \tag {1} \end {equation}
Using Lagrange-charpit
\[ \frac {dx}{a}=\frac {dy}{b}=\frac {dz}{c\tan \left ( \gamma z\right ) }=\frac {dw}{0}\]
From first two pair of equation we obtain \(\frac {b}{a}x-y=C_{1}\) and from \(\frac {dx}{a}=\frac {dz}{c\tan \left ( \gamma z\right ) }\) we obtain \(\frac {c}{a}x-\frac {1}{\gamma }\ln \left ( \sin \left ( \gamma z\right ) \right ) =C_{2}\). Since \(dw=0\) and \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence
\[ u\left ( x,y,z\right ) =F\left ( \frac {b}{a}x-y,\frac {c}{a}x-\frac {1}{\gamma }\ln \left ( \sin \left ( \gamma z\right ) \right ) \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \tan (\beta y) w_y + c \tan (\lambda x) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Tan[beta*y]*D[w[x, y,z], y] +c*Tan[lambda*x]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {c \log (\cos (\lambda x))}{a \lambda }+z,\frac {\log (\sin (\beta y))}{\beta }-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*tan(beta*y)*diff(w(x,y,z),y)+c*tan(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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-b \beta x +a \ln \left (\frac {\tan \left (\beta y \right )}{\sqrt {\tan ^{2}\left (\beta y \right )+1}}\right )}{b \beta }, \frac {2 a \lambda z -c \ln \left (\tan ^{2}\left (\lambda x \right )+1\right )}{2 a \lambda }\right )\]
Hand solution
Solve \begin {equation} aw_{x}+b\tan \left ( \beta y\right ) w_{y}+c\tan \left ( \lambda x\right ) w_{z}=0 \tag {1} \end {equation}
Using Lagrange-charpit
\[ \frac {dx}{a}=\frac {dy}{b\tan \left ( \beta y\right ) }=\frac {dz}{c\tan \left ( \lambda x\right ) }=\frac {dw}{0}\]
From first two pair of equations, integrating gives \(\frac {b}{a}x-\frac {1}{\beta }\ln \left ( \sin \left ( \beta y\right ) \right ) =C_{1}\) and from \(\frac {dx}{a}=\frac {dz}{c\tan \left ( \lambda x\right ) }\) we obtain \(\frac {c}{a}\tan \left ( \lambda x\right ) dx=dz\). Integrating gives \(-\frac {c}{a\lambda }\ln \left ( \cos \left ( \lambda x\right ) \right ) -z=C_{2}\). Since \(dw=0\) then \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence
\[ u\left ( x,y,z\right ) =F\left ( \frac {b}{a}x-\frac {1}{\beta }\ln \left ( \sin \left ( \beta y\right ) \right ) ,-\frac {c}{a\lambda }\ln \left ( \cos \left ( \lambda x\right ) \right ) -z\right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \tan (\beta y) w_y + c \tan (\gamma z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Tan[beta*y]*D[w[x, y,z], y] +c*Tan[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log (\sin (\beta y))}{\beta }-\frac {b x}{a},\frac {b \log \left (\sin ^2(\gamma z)\right )}{\gamma }-\frac {2 c \log (\sin (\beta y))}{\beta }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*tan(beta*y)*diff(w(x,y,z),y)+c*tan(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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-b \beta x +a \ln \left (\frac {\tan \left (\beta y \right )}{\sqrt {\tan ^{2}\left (\beta y \right )+1}}\right )}{b \beta }, \frac {a \ln \left (\frac {\tan \left (\gamma z \right )}{\sqrt {\tan ^{2}\left (\gamma z \right )+1}}\right )-\gamma c x}{\gamma c}\right )\]
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Added May 26, 2019.
Problem Chapter 6.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \tan (\beta y) w_y + c \tan (\gamma z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Tan[beta*y]*D[w[x, y,z], y] +c*Tan[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log (\sin (\beta y))}{\beta }-\frac {b x}{a},\frac {b \log \left (\sin ^2(\gamma z)\right )}{\gamma }-\frac {2 c \log (\sin (\beta y))}{\beta }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*tan(beta*y)*diff(w(x,y,z),y)+c*tan(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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-b \beta x +a \ln \left (\frac {\tan \left (\beta y \right )}{\sqrt {\tan ^{2}\left (\beta y \right )+1}}\right )}{b \beta }, \frac {a \ln \left (\frac {\tan \left (\gamma z \right )}{\sqrt {\tan ^{2}\left (\gamma z \right )+1}}\right )-\gamma c x}{\gamma c}\right )\]
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Added May 26, 2019.
Problem Chapter 6.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ \mu \nu \tan (\lambda x) w_x + \lambda \nu \tan (\mu y) w_y + \lambda \mu \tan (\nu z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = mu*nu*Tan[lambda*x]*D[w[x, y,z], x] + lambda*nu*Tan[mu*y]*D[w[x, y,z], y] +lambda*mu*Tan[nu*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\log \left (\csc ^2(\mu y) \sin ^2(\nu z)\right ),-4 \sin (\lambda x) \csc (\mu y)\right )\right \}\right \}\]
Maple ✓
restart; pde := mu*nu*tan(lambda*x)*diff(w(x,y,z),x)+ lambda*nu*tan(mu*y)*diff(w(x,y,z),y)+lambda*mu*tan(nu*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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\ln \left (\frac {\tan \left (\mu y \right )}{\sqrt {\tan ^{2}\left (\mu y \right )+1}\, \sin \left (\lambda x \right )}\right )}{\lambda }, \frac {\ln \left (\frac {\tan \left (\nu z \right )}{\sqrt {\tan ^{2}\left (\nu z \right )+1}\, \sin \left (\lambda x \right )}\right )}{\lambda }\right )\]
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