Added Oct 18, 2019.
Problem Chapter 8.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \sin ^n(\lambda x) w \]
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*Sin[lambda*x]^n*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
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*sin(lambda*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \left (\sin ^{n}\left (\lambda x \right )\right )d x}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \sin (\lambda z) w_z = \left ( k \sin (\gamma x)+s \sin (\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Sin[lambda*z]*D[w[x,y,z],z]== (k*Sin[gamma*x]+s*Sin[beta*y])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{-\frac {k \cos (\gamma x)}{a \gamma }-\frac {s \cos (\beta y)}{b \beta }} c_1\left (y-\frac {b x}{a},\frac {\log \left (\tan \left (\frac {\lambda z}{2}\right )\right )}{\lambda }-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+ c*sin(lambda*z)*diff(w(x,y,z),z)= (k*sin(gamma*x)+s*sin(beta*y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {a \ln \left (\RootOf \left (\lambda z -\arctan \left (\frac {2 \mathit {\_Z} \,{\mathrm e}^{\frac {c \lambda x}{a}}}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 c \lambda x}{a}}+1}, -\frac {\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 c \lambda x}{a}}-1}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 c \lambda x}{a}}+1}\right )\right )\right )}{c \lambda }\right ) {\mathrm e}^{\frac {-a \gamma s \cos \left (\beta y \right )-b \beta k \cos \left (\gamma x \right )}{a b \beta \gamma }}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta x) w_z = c \sin ^k(\gamma x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*z]^m*D[w[x,y,z],z]== c*Sin[gamma*x]^k*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*z)^m*diff(w(x,y,z),z)= c*sin(gamma*x)^k*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-y +\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x , \int \frac {a \left (\sin ^{-m}\left (\beta z \right )\right )}{b}d z -\left (\int _{}^{y}\left (\sin ^{-n}\left (\lambda \RootOf \left (\mathit {\_b} -y +\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x -\left (\int ^{\mathit {\_Z}}a \left (\sin ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )\right )\right )d\mathit {\_b} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {c \left (\sin ^{k}\left (\gamma \RootOf \left (\mathit {\_b} -y +\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x -\left (\int ^{\mathit {\_Z}}a \left (\sin ^{n}\left (\mathit {\_a} \lambda \right )\right )d \mathit {\_a} \right )\right )\right )\right ) \left (\sin ^{-n}\left (\lambda \RootOf \left (\mathit {\_b} -y +\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x -\left (\int ^{\mathit {\_Z}}a \left (\sin ^{n}\left (\mathit {\_a} \lambda \right )\right )d \mathit {\_a} \right )\right )\right )\right )}{a}d\mathit {\_b}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta y) w_z = \left ( c \sin ^k(\gamma y) + s \sin ^r(\mu z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*y]^m*D[w[x,y,z],z]== (c*Sin[gamma*y]^k+s*Sin[mu*z]^r)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*y)^m*diff(w(x,y,z),z)= (c*sin(gamma*y)^k+s*sin(mu*z)^r)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int _{}^{x}b \left (\sin ^{m}\left (\left (a \left (\int \left (\sin ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )+y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )\right )d\mathit {\_b} \right )\right ) {\mathrm e}^{\int _{}^{x}\left (c \left (-\sin \left (\left (-y -\left (\int a \left (\sin ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )+\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right ) \gamma \right )\right )^{k}+s \left (\sin ^{r}\left (\left (z +\int b \left (\sin ^{m}\left (\left (a \left (\int \left (\sin ^{n}\left (\mathit {\_g} \lambda \right )\right )d \mathit {\_g} \right )+y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )\right )d \mathit {\_g} -\left (\int ^{x}b \left (\sin ^{m}\left (\left (a \left (\int \left (\sin ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )+y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )\right )d \mathit {\_b} \right )\right ) \mu \right )\right )\right )d\mathit {\_g}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \sin (\beta y) w_y + c \sin (\lambda x) w_z = k \sin (\gamma z) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Sin[beta*y]*D[w[x, y,z], y] + c*Sin[lambda*x]^m*D[w[x,y,z],z]== k*Sin[gamma*z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tan \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},z-\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda x)\right )}{a \lambda m+a \lambda }\right ) \exp \left (\int _1^x\frac {k \sin \left (\frac {\gamma \left (-c \sqrt {\cos ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda x)\right ) \sec (\lambda x) \sin ^{m+1}(\lambda x)+c \sqrt {\cos ^2(\lambda K[1])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\lambda K[1])\right ) \sec (\lambda K[1]) \sin ^{m+1}(\lambda K[1])+a \lambda (m+1) z\right )}{a \lambda (m+1)}\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*sin(beta*y)*diff(w(x,y,z),y)+ c*sin(lambda*x)^m*diff(w(x,y,z),z)= k*sin(gamma*z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a \ln \left (\RootOf \left (\beta y -\arctan \left (\frac {2 \mathit {\_Z} \,{\mathrm e}^{\frac {b \beta x}{a}}}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}, -\frac {\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}-1}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}\right )\right )\right )}{b \beta }, z -\left (\int \frac {c \left (\sin ^{m}\left (\lambda x \right )\right )}{a}d x \right )\right ) {\mathrm e}^{\int _{}^{x}-\frac {k \sin \left (\left (-z -\left (\int \frac {c \left (\sin ^{m}\left (\mathit {\_b} \lambda \right )\right )}{a}d \mathit {\_b} \right )+\int \frac {c \left (\sin ^{m}\left (\lambda x \right )\right )}{a}d x \right ) \gamma \right )}{a}d\mathit {\_b}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \sin ^{n_1}(\lambda _1 x) w_x + b_1 \sin ^{m_1}(\beta _1 y) w_y + c_1 \sin ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \sin ^{n_2}(\lambda _2 x) + b_2 \sin ^{m_2}(\beta _2 y) + c_2 \sin ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Sin[lambda1*z]^n1*D[w[x, y,z], x] + b1*Sin[beta1*y]^m1*D[w[x, y,z], y] + c1*Sin[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Sin[lambda2*z]^n2 + b2*Sin[beta2*y]^m2 + c2*Sin[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*sin(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*sin(beta1*y)^m1*diff(w(x,y,z),y)+ c1*sin(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*sin(lambda2*z)^n2 + b2*sin(beta2*y)^m2 + c2*sin(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 ) = \mathit {\_F1} \left (-\left (\int \left (\sin ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\sin ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z , x -\left (\int _{}^{y}\frac {\mathit {a1} \left (\sin ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right ) \left (\sin ^{\mathit {n1}}\left (\lambda 1 \RootOf \left (\int \left (\sin ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\sin ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\sin ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\sin ^{-\mathit {k1}}\left (\mathit {\_b} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_b} \right )\right )\right )\right )}{\mathit {b1}}d\mathit {\_f} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (\mathit {a2} \left (\sin ^{\mathit {n2}}\left (\lambda 2 \RootOf \left (\int \left (\sin ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\sin ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\sin ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\sin ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )\right )+\mathit {b2} \left (\sin ^{\mathit {m2}}\left (\mathit {\_f} \beta 2 \right )\right )+\mathit {c2} \left (\sin ^{\mathit {k2}}\left (\gamma 2 \RootOf \left (\int \left (\sin ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\sin ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\sin ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\sin ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )\right )\right ) \left (\sin ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )}{\mathit {b1}}d\mathit {\_f}}\]
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