Added June 26, 2019.
Problem Chapter 7.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \cot ^{n_1}(\lambda _1 x) w_x + b_1 \cot ^{m_1}(\beta _1 y) w_y + c_1 \cot ^{k_1}(\gamma _1 z) w_z = a_2 \cot ^{n_2}(\lambda _2 x) + b_2 \cot ^{m_2}(\beta _2 y)+ c_2 \cot ^{k_2}(\gamma _2 z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Cot[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cot[beta1*y]^m1*D[w[x, y,z], y] + c1*Cot[gamma1*z]^k1*D[w[x,y,z],z]==a2*Cot[lambda2*z]^n2+ b2*Cot[beta2*y]^m2 + c2*Cot[gamma2*z]^k2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✓
restart; local gamma; pde := a1*cot(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*cot(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cot(gamma1*z)^k1*diff(w(x,y,z),z)= a2*cot(lambda2*x)^n2+ b2*cot(beta2*y)^m2+ c2*cot(gamma2*z)^k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {\left (\mathit {a2} \left (\cot ^{\mathit {n2}}\left (\textit {\_f} \lambda 2 \right )\right )+\mathit {b2} \left (\cot ^{\mathit {m2}}\left (\beta 2 \RootOf \left (\int \left (\cot ^{-\mathit {n1}}\left (\textit {\_f} \lambda 1 \right )\right )d \textit {\_f} -\left (\int \left (\cot ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )}{\mathit {b1}}d y -\left (\int ^{\textit {\_Z}}\frac {\mathit {a1} \left (\cot ^{-\mathit {m1}}\left (\textit {\_a} \beta 1 \right )\right )}{\mathit {b1}}d \textit {\_a} \right )\right )\right )\right )+\mathit {c2} \left (\cot ^{\mathit {k2}}\left (\gamma 2 \RootOf \left (\int \left (\cot ^{-\mathit {n1}}\left (\textit {\_f} \lambda 1 \right )\right )d \textit {\_f} -\left (\int \left (\cot ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\textit {\_Z}}\frac {\mathit {a1} \left (\cot ^{-\mathit {k1}}\left (\textit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \textit {\_a} \right )\right )\right )\right )\right ) \left (\cot ^{-\mathit {n1}}\left (\textit {\_f} \lambda 1 \right )\right )}{\mathit {a1}}d \textit {\_f} +\textit {\_F1} \left (-\left (\int \left (\cot ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )}{\mathit {b1}}d y , -\left (\int \left (\cot ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z \right )\]
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