Added June 26, 2019.
Problem Chapter 7.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \cot ^k(\lambda x)+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] + c*D[w[x,y,z],z]== c*Cot[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)-\frac {c \cot ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*cot(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = s x +\int c \left (\cot ^{k}\left (\lambda x \right )\right )d x +\textit {\_F1} \left (-a x +y , -b x +z \right )\]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \cot (\gamma z) w_z = k \cot (\lambda x)+ s \cot (\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Cot[gamma*z]*D[w[x,y,z],z]== k*Cot[lambda*x]+s*Cot[beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {a \lambda s \log (\tan (\beta y))+a \lambda s \log (\cos (\beta y))+b \beta k \log (\sin (\lambda x))}{a b \beta \lambda }+c_1\left (y-\frac {b x}{a},\frac {\log (\sec (\gamma z))}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*cot(gamma*z)*diff(w(x,y,z),z)= k*cot(lambda*x)+s*cot(beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {2 a b \beta \lambda \textit {\_F1} \left (\frac {-a y +b x}{b}, \frac {-2 c \gamma y +b \ln \left (\cot ^{2}\left (\gamma z \right )+1\right )-2 b \ln \left (\cot \left (\gamma z \right )\right )}{2 c \gamma }\right )-a \lambda s \ln \left (\cot ^{2}\left (\beta y \right )+1\right )-b \beta k \ln \left (\cot ^{2}\left (\lambda x \right )+1\right )}{2 a b \beta \lambda }\]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cot ^n(\beta x) w_y + b \cot ^k(\lambda x) w_z = c \cot ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cot[beta*x]^n*D[w[x, y,z], y] + b*Cot[lambda*x]^k*D[w[x,y,z],z]== c*Cot[gamma*x]^m+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \cot ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }+z,\frac {a \cot ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\beta x)\right )}{\beta n+\beta }+y\right )-\frac {c \cot ^{m+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*cot(beta*x)^n*diff(w(x,y,z),y)+ b*cot(lambda*x)^k*diff(w(x,y,z),z)= c*cot(gamma*x)^m+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = s x +\int c \left (\cot ^{m}\left (\gamma x \right )\right )d x +\textit {\_F1} \left (y -\left (\int a \left (\cot ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int b \left (\cot ^{k}\left (\lambda x \right )\right )d x \right )\right )\]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cot ^n(\lambda x) w_y + b \cot ^m(\beta y) w_z = c \cot ^k(\gamma y)+s \cot ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cot[lambda*x]^n*D[w[x, y,z], y] + b*Cot[beta*x]^m*D[w[x,y,z],z]== c*Cot[gamma*y]^k+s*Cot[mu*z]^r; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \cot ^k\left (\frac {\gamma \left (a \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda x)\right ) \cot ^{n+1}(\lambda x)+\lambda (n+1) y-a \cot ^{n+1}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda K[1])\right )\right )}{\lambda (n+1)}\right )+s \cot ^r\left (\frac {\mu \left (b \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\beta x)\right ) \cot ^{m+1}(\beta x)+\beta (m+1) z-b \cot ^{m+1}(\beta K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\beta K[1])\right )\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (\frac {b \cot ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\beta x)\right )}{\beta m+\beta }+z,\frac {a \cot ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda x)\right )}{\lambda n+\lambda }+y\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*cot(lambda*x)^n*diff(w(x,y,z),y)+ b*cot(beta*x)^m*diff(w(x,y,z),z)= c*cot(gamma*y)^k+s*cot(mu*z)^r; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (c \left (\frac {\cot \left (\left (y -\left (\int a \left (\cot ^{n}\left (\lambda x \right )\right )d x \right )\right ) \gamma \right ) \cot \left (a \gamma \left (\int \left (\cot ^{n}\left (\textit {\_f} \lambda \right )\right )d \textit {\_f} \right )\right )-1}{\cot \left (\left (y -\left (\int a \left (\cot ^{n}\left (\lambda x \right )\right )d x \right )\right ) \gamma \right )+\cot \left (a \gamma \left (\int \left (\cot ^{n}\left (\textit {\_f} \lambda \right )\right )d \textit {\_f} \right )\right )}\right )^{k}+s \left (\frac {\cot \left (\left (z -\left (\int b \left (\cot ^{m}\left (\beta x \right )\right )d x \right )\right ) \mu \right ) \cot \left (b \mu \left (\int \left (\cot ^{m}\left (\textit {\_f} \beta \right )\right )d \textit {\_f} \right )\right )-1}{\cot \left (\left (z -\left (\int b \left (\cot ^{m}\left (\beta x \right )\right )d x \right )\right ) \mu \right )+\cot \left (b \mu \left (\int \left (\cot ^{m}\left (\textit {\_f} \beta \right )\right )d \textit {\_f} \right )\right )}\right )^{r}\right )d \textit {\_f} +\textit {\_F1} \left (y -\left (\int a \left (\cot ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int b \left (\cot ^{m}\left (\beta x \right )\right )d x \right )\right )\]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cot (\beta x) w_y + c \cot (\lambda x) w_z = k \cot (\gamma z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cot[beta*x]*D[w[x, y,z], y] + c*Cot[lambda*x]*D[w[x,y,z],z]== k*Cot[gamma*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \cot \left (\frac {\gamma (a \lambda z-c \log (\sin (\lambda x))+c \log (\sin (\lambda K[1])))}{a \lambda }\right )}{a}dK[1]+c_1\left (y-\frac {b \log (\sin (\beta x))}{a \beta },z-\frac {c \log (\sin (\lambda x))}{a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*cot(beta*x)*diff(w(x,y,z),y)+ c*cot(lambda*x)*diff(w(x,y,z),z)= k*cot(gamma*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {\left (\cot \left (\frac {\left (2 a \lambda z +c \ln \left (\cot ^{2}\left (\lambda x \right )+1\right )\right ) \gamma }{2 a \lambda }\right ) \cot \left (\frac {c \gamma \ln \left (\cot ^{2}\left (\textit {\_a} \lambda \right )+1\right )}{2 a \lambda }\right )+1\right ) k}{\left (-\cot \left (\frac {\left (2 a \lambda z +c \ln \left (\cot ^{2}\left (\lambda x \right )+1\right )\right ) \gamma }{2 a \lambda }\right )+\cot \left (\frac {c \gamma \ln \left (\cot ^{2}\left (\textit {\_a} \lambda \right )+1\right )}{2 a \lambda }\right )\right ) a}d \textit {\_a} +\textit {\_F1} \left (\frac {2 a \beta y +b \ln \left (\cot ^{2}\left (\beta x \right )+1\right )}{2 a \beta }, \frac {2 a \lambda z +c \ln \left (\cot ^{2}\left (\lambda x \right )+1\right )}{2 a \lambda }\right )\]
____________________________________________________________________________________