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]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {i c \sin ^{-1}(\beta x)^n \left (\sin ^{-1}(\beta x)^2\right )^{-n} \left (\left (-i \sin ^{-1}(\beta x)\right )^n \text {Gamma}\left (n+1,i \sin ^{-1}(\beta x)\right )-\left (i \sin ^{-1}(\beta x)\right )^n \text {Gamma}\left (n+1,-i \sin ^{-1}(\beta x)\right )\right )}{2 \beta }\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( \arcsin \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
____________________________________________________________________________________
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya_{1}-xa_{2}}{a_{1}}},{\frac {za_{1}-xa_{3}}{a_{1}}} \right ) {{\rm e}^{{\frac {1}{a_{1}\,\lambda _{1}\,\lambda _{2}\,a_{2}\,\lambda _{3}\,a_{3}} \left ( \lambda _{1}\,\lambda _{3}\,\sqrt {-{y}^{2}{\lambda _{2}}^{2}+1}a_{1}\,a_{3}\,b_{2}+\lambda _{2}\, \left ( \lambda _{1}\,a_{1}\,a_{2}\,b_{3}\,\sqrt {-{z}^{2}{\lambda _{3}}^{2}+1}+ \left ( a_{2}\,a_{3}\,b_{1}\,\sqrt {-{\lambda _{1}}^{2}{x}^{2}+1}+ \left ( x\arcsin \left ( \lambda _{1}\,x \right ) a_{2}\,a_{3}\,b_{1}+a_{1}\, \left ( \arcsin \left ( \lambda _{2}\,y \right ) ya_{3}\,b_{2}+\arcsin \left ( \lambda _{3}\,z \right ) za_{2}\,b_{3} \right ) \right ) \lambda _{1} \right ) \lambda _{3} \right ) \right ) }}}\]
____________________________________________________________________________________
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 ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},-2\,{\frac {1}{ \left ( n+1 \right ) \lambda \, \left ( {\lambda }^{2}{x}^{2}-1 \right ) c\beta \, \left ( k-1 \right ) } \left ( 1/2\,{2}^{n}\lambda \,{2}^{-n}cx\beta \, \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( \lambda \,x \right ) \right ) }{\sqrt {\arcsin \left ( \lambda \,x \right ) }}}+ \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{n} \right ) \left ( -{\lambda }^{2}{x}^{2}+1 \right ) +1/2\,{2}^{n}{2}^{-n}\arcsin \left ( \lambda \,x \right ) c\beta \, \left ( \lambda \,x-1 \right ) \left ( \lambda \,x+1 \right ) \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( \lambda \,x \right ) \right ) }{\sqrt {\arcsin \left ( \lambda \,x \right ) }}}+ \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{n} \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}+ \left ( -1/2\,{\frac {a{2}^{-k}{2}^{k} \left ( n+1 \right ) \left ( \arcsin \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) - \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k} \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{3/2} \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}}{\sqrt {\arcsin \left ( \beta \,z \right ) }}}+\beta \, \left ( 1/2\,{\frac {a{2}^{-k}z{2}^{k} \left ( n+1 \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) }{\sqrt {\arcsin \left ( \beta \,z \right ) }}}-1/2\,a{2}^{-k}\sqrt {\arcsin \left ( \beta \,z \right ) }kz{2}^{k} \left ( n+1 \right ) \LommelS 1 \left ( -k+1/2,3/2,\arcsin \left ( \beta \,z \right ) \right ) +1/2\,{2}^{n}{2}^{-n}c\sqrt {\arcsin \left ( \lambda \,x \right ) }nx \left ( k-1 \right ) \LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( \lambda \,x \right ) \right ) + \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k}az \left ( {2}^{k-1}-1/2\,{2}^{k} \right ) \left ( n+1 \right ) {2}^{-k}+{2}^{n}{2}^{-1-n}c \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{n}x \left ( k-1 \right ) \right ) \right ) \lambda \, \left ( \lambda \,x+1 \right ) \left ( \lambda \,x-1 \right ) \right ) } \right ) {{\rm e}^{{\frac {s}{a\gamma } \left ( \gamma \,x\arcsin \left ( \gamma \,x \right ) +\sqrt {-{\gamma }^{2}{x}^{2}+1} \right ) }}}\]
____________________________________________________________________________________
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},-\int ^{x}\! \left ( \arcsin \left ( \lambda \,{\it \_a} \right ) \right ) ^{n} \left ( \arcsin \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{m}{d{\it \_a}}-{\frac {a{2}^{-k}}{ \left ( k-1 \right ) \gamma \,c} \left ( {2}^{k}\sqrt {-{\gamma }^{2}{z}^{2}+1} \left ( \arcsin \left ( \gamma \,z \right ) \right ) ^{-k+{\frac {3}{2}}}-{2}^{k}\arcsin \left ( \gamma \,z \right ) k\LommelS 1 \left ( -k+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \gamma \,z \right ) \right ) \gamma \,z-{2}^{k}\sqrt {-{\gamma }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \gamma \,z \right ) \right ) \arcsin \left ( \gamma \,z \right ) +{2}^{k}\LommelS 1 \left ( -k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \gamma \,z \right ) \right ) z\gamma \right ) {\frac {1}{\sqrt {\arcsin \left ( \gamma \,z \right ) }}}} \right ) {{\rm e}^{{\frac {xs}{a}}}}\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^z\frac {s \sin ^{-1}(\beta K[3])^{-k} \sin ^{-1}\left (\frac {\gamma \left (c x-a \int _1^z\sin ^{-1}(\beta K[2])^{-k}dK[2]+a \int _1^{K[3]}\sin ^{-1}(\beta K[2])^{-k}dK[2]\right )}{c}\right ){}^m}{c}dK[3]\right ) c_1\left (y-\int _1^x\frac {b \sin ^{-1}(\lambda K[1])^n}{a}dK[1],\int _1^z\sin ^{-1}(\beta K[2])^{-k}dK[2]-\frac {c x}{a}\right )\right \}\right \}\]
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
____________________________________________________________________________________
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 ) ={\it \_F1} \left ( -{\frac {1}{ \left ( n-1 \right ) \lambda \,b} \left ( a{2}^{n}{2}^{-n} \left ( \arcsin \left ( \lambda \,y \right ) \LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,y \right ) \right ) - \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n} \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{{\frac {3}{2}}} \right ) \sqrt {-{\lambda }^{2}{y}^{2}+1}+ \left ( -ay{2}^{n}{2}^{-n}\LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,y \right ) \right ) +a\arcsin \left ( \lambda \,y \right ) \LommelS 1 \left ( -n+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \lambda \,y \right ) \right ) ny{2}^{n}{2}^{-n}+\sqrt {\arcsin \left ( \lambda \,y \right ) } \left ( ay{2}^{-n} \left ( {2}^{n}-2\,{2}^{n-1} \right ) \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n}-bx \left ( n-1 \right ) \right ) \right ) \lambda \right ) {\frac {1}{\sqrt {\arcsin \left ( \lambda \,y \right ) }}}},-2\,{\frac {1}{\lambda \, \left ( n-1 \right ) \left ( {\lambda }^{2}{y}^{2}-1 \right ) c\beta \, \left ( k-1 \right ) } \left ( -1/2\,{2}^{n}\lambda \,{2}^{-n}cy\beta \, \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( \lambda \,y \right ) \right ) }{\sqrt {\arcsin \left ( \lambda \,y \right ) }}}+ \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n} \right ) \left ( -{\lambda }^{2}{y}^{2}+1 \right ) -1/2\,{2}^{n}{2}^{-n}\arcsin \left ( \lambda \,y \right ) c\beta \, \left ( \lambda \,y-1 \right ) \left ( \lambda \,y+1 \right ) \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( \lambda \,y \right ) \right ) }{\sqrt {\arcsin \left ( \lambda \,y \right ) }}}+ \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n} \right ) \sqrt {-{\lambda }^{2}{y}^{2}+1}+ \left ( -1/2\,{\frac {b{2}^{-k}{2}^{k} \left ( n-1 \right ) \left ( \arcsin \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) - \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k} \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{3/2} \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}}{\sqrt {\arcsin \left ( \beta \,z \right ) }}}+\beta \, \left ( 1/2\,{\frac {b{2}^{-k}z{2}^{k} \left ( n-1 \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) }{\sqrt {\arcsin \left ( \beta \,z \right ) }}}+1/2\,{2}^{n}{2}^{-n}c\sqrt {\arcsin \left ( \lambda \,y \right ) }ny \left ( k-1 \right ) \LommelS 1 \left ( -n+1/2,3/2,\arcsin \left ( \lambda \,y \right ) \right ) -1/2\,b{2}^{-k}\sqrt {\arcsin \left ( \beta \,z \right ) }kz{2}^{k} \left ( n-1 \right ) \LommelS 1 \left ( -k+1/2,3/2,\arcsin \left ( \beta \,z \right ) \right ) +bz \left ( {2}^{k-1}-1/2\,{2}^{k} \right ) \left ( n-1 \right ) \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k}{2}^{-k}-{2}^{-n}{2}^{n-1}c \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n}y \left ( k-1 \right ) \right ) \right ) \lambda \, \left ( \lambda \,y+1 \right ) \left ( \lambda \,y-1 \right ) \right ) } \right ) {{\rm e}^{\int \!{\frac {s \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n}}{b}}\,{\rm d}y}}\]
____________________________________________________________________________________