Added Nov 30, 2019.
Problem Chapter 8.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \arcsin ^n(\beta x) w \]
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*ArcSin[beta*x]^n * w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {i c \sin ^{-1}(\beta x)^n \left (\sin ^{-1}(\beta x)^2\right )^{-n} \left (\left (-i \sin ^{-1}(\beta x)\right )^n \operatorname {Gamma}\left (n+1,i \sin ^{-1}(\beta x)\right )-\left (i \sin ^{-1}(\beta x)\right )^n \operatorname {Gamma}\left (n+1,-i \sin ^{-1}(\beta x)\right )\right )}{2 \beta }\right )\right \}\right \}\]
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*arcsin(beta*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \arcsin \left (\beta x \right )^{n}d x}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 w_x + a_2 w_y + a_3 w_z = \left ( b_1 \arcsin (\lambda _1 x)+b_2 \arcsin (\lambda _2 y)+b_3 \arcsin (\lambda _3 z) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a1*D[w[x,y,z],x]+a2*D[w[x,y,z],y]+a3*D[w[x,y,z],z]== (b1*ArcSin[lambda1*x]+b2*ArcSin[lambda2*y]+b3*ArcSin[lambda3*z] ) * w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right ) \exp \left (\frac {\text {b1} \sqrt {1-\text {lambda1}^2 x^2}}{\text {a1} \text {lambda1}}+\frac {\text {b1} x \sin ^{-1}(\text {lambda1} x)}{\text {a1}}+\frac {\text {b2} \sqrt {1-\text {lambda2}^2 y^2}}{\text {a2} \text {lambda2}}+\frac {\text {b2} y \sin ^{-1}(\text {lambda2} y)}{\text {a2}}+\frac {\text {b3} \sqrt {1-\text {lambda3}^2 z^2}}{\text {a3} \text {lambda3}}+\frac {\text {b3} z \sin ^{-1}(\text {lambda3} z)}{\text {a3}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a__1*diff(w(x,y,z),x)+ a__2*diff(w(x,y,z),y)+ a__3*diff(w(x,y,z),z)= (b__1*arcsin(lambda__1*x)+b__2*arcsin(lambda__2*y)+b__3*arcsin(lambda__3*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y a_{1} -x a_{2}}{a_{1}}, \frac {z a_{1} -a_{3} x}{a_{1}}\right ) {\mathrm e}^{\frac {\sqrt {-y^{2} \lambda _{2}^{2}+1}\, a_{1} a_{3} b_{2} \lambda _{1} \lambda _{3} +\left (\sqrt {-z^{2} \lambda _{3}^{2}+1}\, a_{1} a_{2} b_{3} \lambda _{1} +\left (\sqrt {-\lambda _{1}^{2} x^{2}+1}\, a_{2} a_{3} b_{1} +\left (a_{2} a_{3} b_{1} x \arcsin \left (x \lambda _{1} \right )+\left (a_{2} b_{3} z \arcsin \left (z \lambda _{3} \right )+a_{3} b_{2} y \arcsin \left (y \lambda _{2} \right )\right ) a_{1} \right ) \lambda _{1} \right ) \lambda _{3} \right ) \lambda _{2}}{a_{1} a_{2} a_{3} \lambda _{1} \lambda _{2} \lambda _{3}}}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \arcsin ^n(\lambda x) \arcsin ^k(\beta z) w_z = s \arcsin ^m(\gamma x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcSin[lambda*x]^n*ArcSin[beta*z]^k*D[w[x,y,z],z]==s*ArcSin[gamma*x]^m * w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*arcsin(lambda*x)^n*arcsin(beta*z)^k*diff(w(x,y,z),z)= s*arcsin(gamma*x)*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 {2 \left (\frac {\left (k -1\right ) \left (\arcsin \left (\lambda x \right )^{n}-\frac {\LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )}{\sqrt {\arcsin \left (\lambda x \right )}}\right ) \left (-\lambda ^{2} x^{2}+1\right ) \beta c \lambda x 2^{n} 2^{-n}}{2}+\frac {\left (\lambda x -1\right ) \left (\lambda x +1\right ) \left (k -1\right ) \left (\arcsin \left (\lambda x \right )^{n}-\frac {\LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )}{\sqrt {\arcsin \left (\lambda x \right )}}\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, \beta c 2^{n} 2^{-n} \arcsin \left (\lambda x \right )}{2}+\left (\lambda x -1\right ) \left (\lambda x +1\right ) \left (-\frac {\left (n +1\right ) \left (-\arcsin \left (\beta z \right )^{-k} \arcsin \left (\beta z \right )^{\frac {3}{2}}+\LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right ) \arcsin \left (\beta z \right )\right ) \sqrt {-\beta ^{2} z^{2}+1}\, a 2^{k} 2^{-k}}{2 \sqrt {\arcsin \left (\beta z \right )}}+\left (-\frac {\left (n +1\right ) a k z 2^{k} 2^{-k} \LommelS 1\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta z \right )\right ) \sqrt {\arcsin \left (\beta z \right )}}{2}+\frac {\left (k -1\right ) c n x 2^{n} 2^{-n} \LommelS 1\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) \sqrt {\arcsin \left (\lambda x \right )}}{2}+\left (k -1\right ) c x 2^{n} 2^{-n -1} \arcsin \left (\lambda x \right )^{n}+\frac {\left (n +1\right ) a z 2^{k} 2^{-k} \LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right )}{2 \sqrt {\arcsin \left (\beta z \right )}}+\left (-\frac {2^{k}}{2}+2^{k -1}\right ) \left (n +1\right ) a z 2^{-k} \arcsin \left (\beta z \right )^{-k}\right ) \beta \right ) \lambda \right )}{\left (n +1\right ) \left (\lambda ^{2} x^{2}-1\right ) \left (k -1\right ) \beta c \lambda }\right ) {\mathrm e}^{\frac {\left (\gamma x \arcsin \left (\gamma x \right )+\sqrt {-\gamma ^{2} x^{2}+1}\right ) s}{a \gamma }}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \arcsin ^n(\lambda x) \arcsin ^m(\beta y) \arcsin ^k(\gamma z) w_z = s w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcSin[lambda*x]^n*ArcSin[beta*y]^m*ArcSin[gamma*z]^k*D[w[x,y,z],z]==s* w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*arcsin(lambda*x)^n*arcsin(beta*y)^m*arcsin(gamma*z)^k*diff(w(x,y,z),z)= s*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, -\left (\int _{}^{x}\arcsin \left (\mathit {\_a} \lambda \right )^{n} \arcsin \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )^{m}d\mathit {\_a} \right )-\frac {\left (-\gamma k z 2^{k} \LommelS 1\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\gamma z \right )\right ) \arcsin \left (\gamma z \right )+\gamma z 2^{k} \LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\gamma z \right )\right )-\sqrt {-\gamma ^{2} z^{2}+1}\, 2^{k} \LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\gamma z \right )\right ) \arcsin \left (\gamma z \right )+\sqrt {-\gamma ^{2} z^{2}+1}\, 2^{k} \arcsin \left (\gamma z \right )^{-k +\frac {3}{2}}\right ) a 2^{-k}}{\left (k -1\right ) c \gamma \sqrt {\arcsin \left (\gamma z \right )}}\right ) {\mathrm e}^{\frac {s x}{a}}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \arcsin ^n(\lambda x) w_y + c \arcsin ^k(\beta z) w_z = s \arcsin ^m(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*ArcSin[lambda*x]^n*D[w[x,y,z],y]+c*ArcSin[beta*z]^k*D[w[x,y,z],z]==s* ArcSin[gamma*x]^m*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}-\frac {i \sin ^{-1}(\beta z)^{-k} \left (\left (-i \sin ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \sin ^{-1}(\beta z)\right )-\left (i \sin ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \sin ^{-1}(\beta z)\right )\right )}{2 \beta },\frac {\left (\sin ^{-1}(\lambda x)^2\right )^{-n} \left (i b \left (i \sin ^{-1}(\lambda x)\right )^n \sin ^{-1}(\lambda x)^n \operatorname {Gamma}\left (n+1,-i \sin ^{-1}(\lambda x)\right )-i b \left (-i \sin ^{-1}(\lambda x)\right )^n \sin ^{-1}(\lambda x)^n \operatorname {Gamma}\left (n+1,i \sin ^{-1}(\lambda x)\right )+2 a \lambda y \left (\sin ^{-1}(\lambda x)^2\right )^n\right )}{2 a \lambda }\right ) \exp \left (\int _1^z\frac {s \sin ^{-1}\left (\frac {\gamma \left (i a \left (-i \sin ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \sin ^{-1}(\beta z)\right ) \sin ^{-1}(\beta z)^{-k}-i a \left (i \sin ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \sin ^{-1}(\beta z)\right ) \sin ^{-1}(\beta z)^{-k}+\sin ^{-1}(\beta K[1])^{-k} \left (-i a \operatorname {Gamma}\left (1-k,-i \sin ^{-1}(\beta K[1])\right ) \left (-i \sin ^{-1}(\beta K[1])\right )^k+2 \beta c x \sin ^{-1}(\beta K[1])^k+i a \left (i \sin ^{-1}(\beta K[1])\right )^k \operatorname {Gamma}\left (1-k,i \sin ^{-1}(\beta K[1])\right )\right )\right )}{2 \beta c}\right )^m \sin ^{-1}(\beta K[1])^{-k}}{c}dK[1]\right )\right \}\right \}\]
Maple ✗
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*arcsin(lambda*x)^n*diff(w(x,y,z),y)+ c*arcsin(beta*z)^k*diff(w(x,y,z),z)= s*arcsin(gamma*x)^m*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
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Added Nov 30, 2019.
Problem Chapter 8.7.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \arcsin ^n(\lambda y) w_y + c \arcsin ^k(\beta z) w_z = s w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*ArcSin[lambda*y]^n*D[w[x,y,z],y]+c*ArcSin[beta*z]^k*D[w[x,y,z],z]==s* w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*arcsin(lambda*y)^n*diff(w(x,y,z),y)+ c*arcsin(beta*z)^k*diff(w(x,y,z),z)= s*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 {\left (-\arcsin \left (\lambda y \right )^{-n} \arcsin \left (\lambda y \right )^{\frac {3}{2}}+\LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )\right ) \sqrt {-\lambda ^{2} y^{2}+1}\, a 2^{n} 2^{-n}+\left (a n y 2^{n} 2^{-n} \LommelS 1\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )-a y 2^{n} 2^{-n} \LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )+\left (\left (2^{n}-2 2^{n -1}\right ) a y 2^{-n} \arcsin \left (\lambda y \right )^{-n}-\left (n -1\right ) b x \right ) \sqrt {\arcsin \left (\lambda y \right )}\right ) \lambda }{\left (n -1\right ) b \lambda \sqrt {\arcsin \left (\lambda y \right )}}, -\frac {2 \left (-\frac {\left (k -1\right ) \left (\arcsin \left (\lambda y \right )^{-n}-\frac {\LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )}{\sqrt {\arcsin \left (\lambda y \right )}}\right ) \left (-\lambda ^{2} y^{2}+1\right ) \beta c \lambda y 2^{n} 2^{-n}}{2}-\frac {\left (\lambda y -1\right ) \left (\lambda y +1\right ) \left (k -1\right ) \left (\arcsin \left (\lambda y \right )^{-n}-\frac {\LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )}{\sqrt {\arcsin \left (\lambda y \right )}}\right ) \sqrt {-\lambda ^{2} y^{2}+1}\, \beta c 2^{n} 2^{-n} \arcsin \left (\lambda y \right )}{2}+\left (\lambda y +1\right ) \left (-\frac {\left (n -1\right ) \left (-\arcsin \left (\beta z \right )^{-k} \arcsin \left (\beta z \right )^{\frac {3}{2}}+\LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right ) \arcsin \left (\beta z \right )\right ) \sqrt {-\beta ^{2} z^{2}+1}\, b 2^{k} 2^{-k}}{2 \sqrt {\arcsin \left (\beta z \right )}}+\left (-\frac {\left (n -1\right ) b k z 2^{k} 2^{-k} \LommelS 1\left (-k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\beta z \right )\right ) \sqrt {\arcsin \left (\beta z \right )}}{2}+\frac {\left (k -1\right ) c n y 2^{n} 2^{-n} \LommelS 1\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda y \right )\right ) \sqrt {\arcsin \left (\lambda y \right )}}{2}-\left (k -1\right ) c y 2^{-n} 2^{n -1} \arcsin \left (\lambda y \right )^{-n}+\frac {\left (n -1\right ) b z 2^{k} 2^{-k} \LommelS 1\left (-k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\beta z \right )\right )}{2 \sqrt {\arcsin \left (\beta z \right )}}+\left (n -1\right ) \left (-\frac {2^{k}}{2}+2^{k -1}\right ) b z 2^{-k} \arcsin \left (\beta z \right )^{-k}\right ) \beta \right ) \left (\lambda y -1\right ) \lambda \right )}{\left (n -1\right ) \left (\lambda ^{2} y^{2}-1\right ) \left (k -1\right ) \beta c \lambda }\right ) {\mathrm e}^{\int \frac {s \arcsin \left (\lambda y \right )^{-n}}{b}d y}\]
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