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]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \tan ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \left (\tan ^{n}\left (\beta x \right )\right )d x}\]
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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'));
\[w \left (x , y , z\right ) = \left (\tan ^{2}\left (\gamma y \right )+1\right )^{\frac {s}{2 b \gamma }} \left (\tan ^{2}\left (\lambda x \right )+1\right )^{\frac {k}{2 a \lambda }} \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {-\beta c x +a \ln \left (\frac {\tan \left (\beta z \right )}{\sqrt {\tan ^{2}\left (\beta z \right )+1}}\right )}{\beta c}\right )\]
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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]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tan ^{m+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (y-\frac {a \tan ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tan ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\tan ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (y -\left (\int a \left (\tan ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int b \left (\tan ^{k}\left (\lambda x \right )\right )d x \right )\right ) {\mathrm e}^{\int c \left (\tan ^{m}\left (\gamma x \right )\right )d x}\]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log (\sin (\beta y))}{\beta }-\frac {b x}{a},\frac {-c \tan ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\lambda x)\right )+a \lambda m z+a \lambda z}{a \lambda m+a \lambda }\right ) \exp \left (\int _1^x\frac {k \tan \left (\frac {\gamma \left (-c \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\lambda x)\right ) \tan ^{m+1}(\lambda x)+c \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\lambda K[1])\right ) \tan ^{m+1}(\lambda K[1])+a \lambda (m+1) z\right )}{a \lambda (m+1)}\right )}{a}dK[1]\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {-b \beta x +a \ln \left (\frac {\tan \left (\beta y \right )}{\sqrt {\tan ^{2}\left (\beta y \right )+1}}\right )}{b \beta }, z -\left (\int \frac {c \left (\tan ^{m}\left (\lambda x \right )\right )}{a}d x \right )\right ) {\mathrm e}^{\int _{}^{x}-\frac {k \tan \left (\left (-z -\left (\int \frac {c \left (\tan ^{m}\left (\textit {\_b} \lambda \right )\right )}{a}d \textit {\_b} \right )+\int \frac {c \left (\tan ^{m}\left (\lambda x \right )\right )}{a}d x \right ) \gamma \right )}{a}d \textit {\_b}}\]
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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 ) = \textit {\_F1} \left (-\left (\int \left (\tan ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\tan ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z , x -\left (\int _{}^{y}\frac {\mathit {a1} \left (\tan ^{-\mathit {m1}}\left (\textit {\_f} \beta 1 \right )\right ) \left (\tan ^{\mathit {n1}}\left (\lambda 1 \RootOf \left (\int \left (\tan ^{-\mathit {m1}}\left (\textit {\_f} \beta 1 \right )\right )d \textit {\_f} -\left (\int \left (\tan ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\tan ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\textit {\_Z}}\frac {\mathit {b1} \left (\tan ^{-\mathit {k1}}\left (\textit {\_b} \gamma 1 \right )\right )}{\mathit {c1}}d \textit {\_b} \right )\right )\right )\right )}{\mathit {b1}}d \textit {\_f} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (\mathit {a2} \left (\tan ^{\mathit {n2}}\left (\lambda 2 \RootOf \left (\int \left (\tan ^{-\mathit {m1}}\left (\textit {\_f} \beta 1 \right )\right )d \textit {\_f} -\left (\int \left (\tan ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\tan ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\textit {\_Z}}\frac {\mathit {b1} \left (\tan ^{-\mathit {k1}}\left (\textit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \textit {\_a} \right )\right )\right )\right )+\mathit {b2} \left (\tan ^{\mathit {m2}}\left (\textit {\_f} \beta 2 \right )\right )+\mathit {c2} \left (\tan ^{\mathit {k2}}\left (\gamma 2 \RootOf \left (\int \left (\tan ^{-\mathit {m1}}\left (\textit {\_f} \beta 1 \right )\right )d \textit {\_f} -\left (\int \left (\tan ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\tan ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\textit {\_Z}}\frac {\mathit {b1} \left (\tan ^{-\mathit {k1}}\left (\textit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \textit {\_a} \right )\right )\right )\right )\right ) \left (\tan ^{-\mathit {m1}}\left (\textit {\_f} \beta 1 \right )\right )}{\mathit {b1}}d \textit {\_f}}\]
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