Added Oct 18, 2019.
Problem Chapter 8.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \tan ^n(\beta x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] + b*D[w[x,y,z],z]== c*Tan[beta*x]^n*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tan ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\beta x)\right )}{\beta n+\beta }\right ) c_1(y-a x,z-b x)\right \}\right \}\]
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(beta*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( \tan \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \tan (\beta z) w_z = \left ( k \tan (\lambda x)+s \tan (\gamma y) \right ) w \]
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])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \cos ^{-\frac {k}{a \lambda }}(\lambda x) \cos ^{-\frac {s}{b \gamma }}(\gamma y) c_1\left (y-\frac {b x}{a},\frac {\log (\sin (\beta z))}{\beta }-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+ c*tan(beta*z)*diff(w(x,y,z),z)= (k*tan(lambda*x)+s*tan(gamma*y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {1}{c\beta } \left ( -xc\beta +\ln \left ( {\tan \left ( \beta \,z \right ) {\frac {1}{\sqrt {1+ \left ( \tan \left ( \beta \,z \right ) \right ) ^{2}}}}} \right ) a \right ) } \right ) \left ( 1+ \left ( \tan \left ( \lambda \,x \right ) \right ) ^{2} \right ) ^{{\frac {k}{2\,a\lambda }}} \left ( 1+ \left ( \tan \left ( \gamma \,y \right ) \right ) ^{2} \right ) ^{{\frac {s}{2\,\gamma \,b}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \tan ^n(\beta x) w_y + b \tan ^k(\lambda x) w_z = c \tan ^m(\gamma x) w \]
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*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tan ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (y-\frac {a \tan ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tan ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\tan ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
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*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \tan \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \tan \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) {{\rm e}^{\int \!c \left ( \tan \left ( \gamma \,x \right ) \right ) ^{m}\,{\rm d}x}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \tan (\beta y) w_y + c \tan (\lambda x) w_z = k \tan (\gamma z) w \]
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]^m*D[w[x,y,z],z]== k*Tan[gamma*z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log (\sin (\beta y))}{\beta }-\frac {b x}{a},\frac {a \lambda m z+a \lambda z-c \tan ^{m+1}(\lambda x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(\lambda x)\right )}{a \lambda m+a \lambda }\right ) \exp \left (\int _1^x\frac {k \tan \left (\frac {\gamma \left (-c \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(\lambda x)\right ) \tan ^{m+1}(\lambda x)+c \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(\lambda K[1])\right ) \tan ^{m+1}(\lambda K[1])+a \lambda (m+1) z\right )}{a \lambda (m+1)}\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*tan(beta*y)*diff(w(x,y,z),y)+ c*tan(lambda*x)^m*diff(w(x,y,z),z)= k*tan(gamma*z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{b\beta } \left ( -xb\beta +\ln \left ( {\tan \left ( \beta \,y \right ) {\frac {1}{\sqrt {1+ \left ( \tan \left ( \beta \,y \right ) \right ) ^{2}}}}} \right ) a \right ) },-\int \!{\frac {c \left ( \tan \left ( \lambda \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+z \right ) {{\rm e}^{\int ^{x}\!-{\frac {k}{a}\tan \left ( \gamma \, \left ( -\int \!{\frac {c \left ( \tan \left ( \lambda \,{\it \_b} \right ) \right ) ^{m}}{a}}\,{\rm d}{\it \_b}+\int \!{\frac {c \left ( \tan \left ( \lambda \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-z \right ) \right ) }{d{\it \_b}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \tan ^{n_1}(\lambda _1 x) w_x + b_1 \tan ^{m_1}(\beta _1 y) w_y + c_1 \tan ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \tan ^{n_2}(\lambda _2 x) + b_2 \tan ^{m_2}(\beta _2 y) + c_2 \tan ^{k_2}(\gamma _2 z) \right ) w \]
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)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*tan(lambda1*z)^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*z)^n2 + b2*tan(beta2*y)^m2 + c2*tan(gamma2*z)^k2)*w(x,y,z); 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 ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \tan \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z,-\int ^{y}\!{\frac {{\it a1}\, \left ( \tan \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( \tan \left ( \lambda 1\,\RootOf \left ( \int \! \left ( \tan \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \tan \left ( \gamma 1\,{\it \_b} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_b}}-\int \! \left ( \tan \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \tan \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n1}}}{d{\it \_f}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \tan \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( {\it a2}\, \left ( \tan \left ( \lambda 2\,\RootOf \left ( \int \! \left ( \tan \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \tan \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \tan \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \tan \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n2}}+{\it c2}\, \left ( \tan \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \tan \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \tan \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \tan \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \tan \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}}+{\it b2}\, \left ( \tan \left ( \beta 2\,{\it \_f} \right ) \right ) ^{{\it m2}} \right ) }{d{\it \_f}}}}\]
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