Problem Chapter 7.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
\[ a_1 \cot ^{n_1}(\lambda _1 x) w_x + b_1 \cot ^{m_1}(\beta _1 y) w_y + c_1 \cot ^{k_1}(\gamma _1 z) w_z = a_2 \cot ^{n_2}(\lambda _2 x) + b_2 \cot ^{m_2}(\beta _2 y)+ c_2 \cot ^{k_2}(\gamma _2 z) \]
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]];
Failed
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'));
\[w \left (x , y , z\right ) = f_{1} \left (-\int \cot \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y}{\operatorname {b1}}, -\int \cot \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \cot \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z}{\operatorname {c1}}\right )+\frac {\int _{}^{x}\cot \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}} \left (\operatorname {a2} \cot \left (\lambda \operatorname {2} \textit {\_f} \right )^{\operatorname {n2}}+\operatorname {b2} {\cot \left (\beta \operatorname {2} \operatorname {RootOf}\left (\operatorname {a1} \int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y -\operatorname {a1} \int _{}^{\textit {\_Z}}\cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} +\int \cot \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {b1} -\int \cot \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {b1} \right )\right )}^{\operatorname {m2}}+\operatorname {c2} {\cot \left (\gamma \operatorname {2} \operatorname {RootOf}\left (\int \cot \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {c1} -\operatorname {a1} \int _{}^{\textit {\_Z}}\cot \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} -\int \cot \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {c1} +\operatorname {a1} \int \cot \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z \right )\right )}^{\operatorname {k2}}\right )d \textit {\_f}}{\operatorname {a1}}\]
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