Added June 26, 2019.
Problem Chapter 7.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \tan ^k(\lambda x)+s \]
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*Tan[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)+\frac {c \tan ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\tan ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*tan(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \tan \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]
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Added June 26, 2019.
Problem Chapter 7.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \tan (\beta z) w_z = k \tan (\lambda x)+ s \tan (\gamma y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Tan[beta*z]*D[w[x,y,z],z]== k*Tan[lambda*x]+s*Tan[gamma*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log (\sin (\beta z))}{\beta }-\frac {c x}{a}\right )-\frac {k \log (\cos (\lambda x))}{a \lambda }-\frac {s \log (\cos (\gamma y))}{b \gamma }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*tan(beta*z)*diff(w(x,y,z),z)= k*tan(lambda*x)+s*tan(gamma*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{2\,b\gamma \,a\lambda } \left ( 2\,{\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {1}{\beta \,c} \left ( -x\beta \,c+\ln \left ( {\frac {\tan \left ( \beta \,z \right ) }{\sqrt {1+ \left ( \tan \left ( \beta \,z \right ) \right ) ^{2}}}} \right ) a \right ) } \right ) b\gamma \,a\lambda +k\ln \left ( 1+ \left ( \tan \left ( x\lambda \right ) \right ) ^{2} \right ) b\gamma +s\ln \left ( 1+ \left ( \tan \left ( \gamma \,y \right ) \right ) ^{2} \right ) a\lambda \right ) }\]
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Added June 26, 2019.
Problem Chapter 7.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \tan ^n(\beta x) w_y + b \tan ^k(\lambda x) w_z = c \tan ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Tan[beta*x]^n*D[w[x, y,z], y] + b*Tan[lambda*x]^k*D[w[x,y,z],z]== c*Tan[gamma*x]^m+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a \tan ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tan ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\tan ^2(\lambda x)\right )}{k \lambda +\lambda }\right )+\frac {c \tan ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*tan(beta*x)^n*diff(w(x,y,z),y)+ b*tan(lambda*x)^k*diff(w(x,y,z),z)= c*tan(gamma*x)^m+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \tan \left ( x\gamma \right ) \right ) ^{m}\,{\rm d}x+sx+{\it \_F1} \left ( -\int \!a \left ( \tan \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \tan \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) \]
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Added June 26, 2019.
Problem Chapter 7.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \tan ^n(\lambda x) w_y + b \tan ^m(\beta y) w_z = c \tan ^k(\gamma y)+s \tan ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Tan[lambda*x]^n*D[w[x, y,z], y] + b*Tan[beta*x]^m*D[w[x,y,z],z]== c*Tan[gamma*y]^k+s*Tan[mu*z]^r; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \tan ^k\left (\frac {\gamma \left (-a \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\lambda x)\right ) \tan ^{n+1}(\lambda x)+a \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\lambda K[1])\right ) \tan ^{n+1}(\lambda K[1])+\lambda (n+1) y\right )}{\lambda (n+1)}\right )+s \tan ^r\left (\frac {\mu \left (-b \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(\beta x)\right ) \tan ^{m+1}(\beta x)+b \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(\beta K[1])\right ) \tan ^{m+1}(\beta K[1])+\beta (m+1) z\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (z-\frac {b \tan ^{m+1}(\beta x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \tan ^{n+1}(\lambda x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*tan(lambda*x)^n*diff(w(x,y,z),y)+ b*tan(beta*x)^m*diff(w(x,y,z),z)= c*tan(gamma*y)^k+s*tan(mu*z)^r; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!c \left ( {\frac {\tan \left ( \gamma \, \left ( -\int \!a \left ( \tan \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) +\tan \left ( \gamma \,a\int \! \left ( \tan \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f} \right ) }{1-\tan \left ( \gamma \, \left ( -\int \!a \left ( \tan \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \tan \left ( \gamma \,a\int \! \left ( \tan \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f} \right ) }} \right ) ^{k}+s \left ( {\frac {\tan \left ( \mu \, \left ( -\int \!b \left ( \tan \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \right ) +\tan \left ( \mu \,b\int \! \left ( \tan \left ( \beta \,{\it \_f} \right ) \right ) ^{m}\,{\rm d}{\it \_f} \right ) }{1-\tan \left ( \mu \, \left ( -\int \!b \left ( \tan \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \right ) \tan \left ( \mu \,b\int \! \left ( \tan \left ( \beta \,{\it \_f} \right ) \right ) ^{m}\,{\rm d}{\it \_f} \right ) }} \right ) ^{r}{d{\it \_f}}+{\it \_F1} \left ( -\int \!a \left ( \tan \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \tan \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \]
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Added June 26, 2019.
Problem Chapter 7.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \tan ^{n_1}(\lambda _1 x) w_x + b_1 \tan ^{m_1}(\beta _1 y) w_y + c_1 \tan ^{k_1}(\gamma _1 z) w_z = a_2 \tan ^{n_2}(\lambda _2 x) + b_2 \tan ^{m_2}(\beta _2 y)+ c_2 \tan ^{k_2}(\gamma _2 z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Tan[lambda1*z]^n1*D[w[x, y,z], x] + b1*Tan[beta1*y]^m1*D[w[x, y,z], y] + c1*Tan[gamma1*z]^k1*D[w[x,y,z],z]==a2*Tan[lambda2*z]^n2+ b2*Tan[beta2*y]^m2 + c2*Tan[gamma2*z]^k2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*tan(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*tan(beta1*y)^m1*diff(w(x,y,z),y)+ c1*tan(gamma1*z)^k1*diff(w(x,y,z),z)= a2*tan(lambda2*x)^n2+ b2*tan(beta2*y)^m2+ c2*tan(gamma2*z)^k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( \tan \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \tan \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y,-\int \! \left ( \tan \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \tan \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) +\int ^{x}\!{\frac { \left ( \tan \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( {\it a2}\, \left ( \tan \left ( \lambda 2\,{\it \_f} \right ) \right ) ^{{\it n2}}+ \left ( \tan \left ( \beta 2\,\RootOf \left ( \int \! \left ( \tan \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \tan \left ( \beta 1\,{\it \_a} \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}{d{\it \_a}}-\int \! \left ( \tan \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \tan \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y \right ) \right ) \right ) ^{{\it m2}}{\it b2}+ \left ( \tan \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \tan \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \tan \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \tan \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \tan \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}}{\it c2} \right ) }{d{\it \_f}}\]
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