Added Nov 30, 2019.
Problem Chapter 8.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \arccos ^n(\beta x) w \]
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*ArcCos[beta*x]^n * w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \cos ^{-1}(\beta x)^n \left (\cos ^{-1}(\beta x)^2\right )^{-n} \left (\left (-i \cos ^{-1}(\beta x)\right )^n \text {Gamma}\left (n+1,i \cos ^{-1}(\beta x)\right )+\left (i \cos ^{-1}(\beta x)\right )^n \text {Gamma}\left (n+1,-i \cos ^{-1}(\beta x)\right )\right )}{2 \beta }\right )\right \}\right \}\]
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*arccos(beta*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( \arccos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
____________________________________________________________________________________
Added Nov 30, 2019.
Problem Chapter 8.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 w_x + a_2 w_y + a_3 w_z = \left ( b_1 \arccos (\lambda _1 x)+b_2 \arccos (\lambda _2 y)+b_3 \arccos (\lambda _3 z) \right ) w \]
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*ArcCos[lambda1*x]+b2*ArcCos[lambda2*y]+b3*ArcCos[lambda3*z] ) * w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right ) \exp \left (-\frac {\text {b1} \sqrt {1-\text {lambda1}^2 x^2}}{\text {a1} \text {lambda1}}+\frac {\text {b1} x \cos ^{-1}(\text {lambda1} x)}{\text {a1}}+\frac {\text {b2} x \sin ^{-1}(\text {lambda2} y)}{\text {a1}}+\frac {\text {b2} x \cos ^{-1}(\text {lambda2} y)}{\text {a1}}+\frac {\text {b3} x \sin ^{-1}(\text {lambda3} z)}{\text {a1}}+\frac {\text {b3} x \cos ^{-1}(\text {lambda3} z)}{\text {a1}}-\frac {\text {b2} \sqrt {1-\text {lambda2}^2 y^2}}{\text {a2} \text {lambda2}}-\frac {\text {b2} y \sin ^{-1}(\text {lambda2} y)}{\text {a2}}-\frac {\text {b3} \sqrt {1-\text {lambda3}^2 z^2}}{\text {a3} \text {lambda3}}-\frac {\text {b3} z \sin ^{-1}(\text {lambda3} z)}{\text {a3}}\right )\right \}\right \}\]
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*arccos(lambda__1*x)+b__2*arccos(lambda__2*y)+b__3*arccos(lambda__3*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya_{1}-a_{2}\,x}{a_{1}}},{\frac {za_{1}-a_{3}\,x}{a_{1}}} \right ) {{\rm e}^{{\frac {-\lambda _{1}\,\lambda _{3}\,\sqrt {-{y}^{2}{\lambda _{2}}^{2}+1}a_{1}\,a_{3}\,b_{2}+\lambda _{2}\, \left ( -\lambda _{1}\,a_{1}\,a_{2}\,b_{3}\,\sqrt {-{z}^{2}{\lambda _{3}}^{2}+1}+ \left ( -a_{2}\,a_{3}\,b_{1}\,\sqrt {-{\lambda _{1}}^{2}{x}^{2}+1}+ \left ( x\arccos \left ( \lambda _{1}\,x \right ) a_{2}\,a_{3}\,b_{1}+a_{1}\, \left ( \arccos \left ( \lambda _{2}\,y \right ) ya_{3}\,b_{2}+\arccos \left ( \lambda _{3}\,z \right ) za_{2}\,b_{3} \right ) \right ) \lambda _{1} \right ) \lambda _{3} \right ) }{a_{1}\,\lambda _{1}\,\lambda _{2}\,a_{2}\,\lambda _{3}\,a_{3}}}}}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \arccos ^n(\lambda x) \arccos ^k(\beta z) w_z = s \arccos ^m(\gamma x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcCos[lambda*x]^n*ArcCos[beta*z]^k*D[w[x,y,z],z]==s*ArcCos[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*arccos(lambda*x)^n*arccos(beta*z)^k*diff(w(x,y,z),z)= s*arccos(gamma*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-bx}{a}},{\frac {{2}^{n}\sqrt {\pi }}{\lambda } \left ( {\frac { \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+1}{2}^{-n}\sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( n+2 \right ) }}-{\frac {{2}^{-n}\sqrt {\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( n+2 \right ) }}-3\,{\frac {{2}^{-1-n} \left ( 2/3\,n+4/3 \right ) \left ( \lambda \,x\arccos \left ( \lambda \,x \right ) -\sqrt {-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt {\pi } \left ( n+2 \right ) \sqrt {\arccos \left ( \lambda \,x \right ) }}} \right ) }-{\frac {a{2}^{-k} \left ( -2\,{2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\arccos \left ( \beta \,z \right ) \beta \,k+4\,{2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\arccos \left ( \beta \,z \right ) \beta +{2}^{k}\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,3/2,\arccos \left ( \beta \,z \right ) \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}+2\,{2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) k- \left ( \arccos \left ( \beta \,z \right ) \right ) ^{-k+1}{2}^{k}\sqrt {-{\beta }^{2}{z}^{2}+1}\sqrt {\arccos \left ( \beta \,z \right ) }-4\,{2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) \right ) }{ \left ( k-2 \right ) c\beta \,\sqrt {\arccos \left ( \beta \,z \right ) }}} \right ) {{\rm e}^{{\frac {s \left ( \gamma \,x\arccos \left ( \gamma \,x \right ) -\sqrt {-{\gamma }^{2}{x}^{2}+1} \right ) }{a\gamma }}}}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \arccos ^n(\lambda x) \arccos ^m(\beta y) \arccos ^k(\gamma z) w_z = s w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcCos[lambda*x]^n*ArcCos[beta*y]^m*ArcCos[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*arccos(lambda*x)^n*arccos(beta*y)^m*arccos(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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-bx}{a}},{\frac {1}{ \left ( k-2 \right ) \gamma \,c} \left ( -\int ^{x}\! \left ( \arccos \left ( \lambda \,{\it \_a} \right ) \right ) ^{n} \left ( \arccos \left ( {\frac { \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \beta }{a}} \right ) \right ) ^{m}{d{\it \_a}}c\gamma \, \left ( k-2 \right ) +{\frac { \left ( \left ( \left ( -k+2 \right ) \LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \gamma \,z \right ) \right ) -\arccos \left ( \gamma \,z \right ) \LommelS 1 \left ( -k+3/2,3/2,\arccos \left ( \gamma \,z \right ) \right ) + \left ( \arccos \left ( \gamma \,z \right ) \right ) ^{-k+3/2} \right ) \sqrt {-{\gamma }^{2}{z}^{2}+1}+\gamma \,\arccos \left ( \gamma \,z \right ) \LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \gamma \,z \right ) \right ) z \left ( k-2 \right ) \right ) {2}^{-k}{2}^{k}a}{\sqrt {\arccos \left ( \gamma \,z \right ) }}} \right ) } \right ) {{\rm e}^{{\frac {sx}{a}}}}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \arccos ^n(\lambda x) w_y + c \arccos ^k(\beta z) w_z = s \arccos ^m(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*ArcCos[lambda*x]^n*D[w[x,y,z],y]+c*ArcCos[beta*z]^k*D[w[x,y,z],z]==s* ArcCos[gamma*x]^m*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta },\frac {\left (\cos ^{-1}(\lambda x)^2\right )^{-n} \left (-b \left (i \cos ^{-1}(\lambda x)\right )^n \cos ^{-1}(\lambda x)^n \text {Gamma}\left (n+1,-i \cos ^{-1}(\lambda x)\right )-b \left (-i \cos ^{-1}(\lambda x)\right )^n \cos ^{-1}(\lambda x)^n \text {Gamma}\left (n+1,i \cos ^{-1}(\lambda x)\right )+2 a \lambda y \left (\cos ^{-1}(\lambda x)^2\right )^n\right )}{2 a \lambda }\right ) \exp \left (\int _1^z\frac {s \cos ^{-1}\left (\frac {\gamma \left (-a \left (-i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right ) \cos ^{-1}(\beta z)^{-k}-a \left (i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right ) \cos ^{-1}(\beta z)^{-k}+\cos ^{-1}(\beta K[1])^{-k} \left (a \text {Gamma}\left (1-k,-i \cos ^{-1}(\beta K[1])\right ) \left (-i \cos ^{-1}(\beta K[1])\right )^k+2 \beta c x \cos ^{-1}(\beta K[1])^k+a \left (i \cos ^{-1}(\beta K[1])\right )^k \text {Gamma}\left (1-k,i \cos ^{-1}(\beta K[1])\right )\right )\right )}{2 \beta c}\right )^m \cos ^{-1}(\beta K[1])^{-k}}{c}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*arccos(lambda*x)^n*diff(w(x,y,z),y)+ c*arccos(beta*z)^k*diff(w(x,y,z),z)= s*arccos(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 ( {\frac {1}{a\lambda \, \left ( n+2 \right ) } \left ( -{\frac { \left ( \left ( n+2 \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) -\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) \arccos \left ( \lambda \,x \right ) + \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+3/2} \right ) {2}^{-n}{2}^{n}b\sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\arccos \left ( \lambda \,x \right ) }}}-\lambda \, \left ( n+2 \right ) \left ( -\sqrt {\arccos \left ( \lambda \,x \right ) }bx{2}^{n}{2}^{-n}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) +ay \right ) \right ) },{\frac {1}{c\beta \, \left ( k-2 \right ) } \left ( -\int ^{y}\! \left ( \arccos \left ( \lambda \,\RootOf \left ( \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \arccos \left ( \lambda \,{\it \_Z} \right ) {\it \_Z}\,b\lambda \,n-a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }n\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-{\it \_b}\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }n+a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }ny+2\,\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \arccos \left ( \lambda \,{\it \_Z} \right ) {\it \_Z}\,b\lambda -2\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-2\,{\it \_b}\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }+2\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }y+\arccos \left ( \lambda \,{\it \_Z} \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b-\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}bn- \left ( \arccos \left ( \lambda \,{\it \_Z} \right ) \right ) ^{n+3/2}b\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}-2\,\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b \right ) \right ) \right ) ^{-n}{d{\it \_b}}c\beta \, \left ( k-2 \right ) +{\frac {{2}^{k}{2}^{-k} \left ( \left ( \left ( -k+2 \right ) \LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) -\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,3/2,\arccos \left ( \beta \,z \right ) \right ) + \left ( \arccos \left ( \beta \,z \right ) \right ) ^{-k+3/2} \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}+\beta \,\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z \left ( k-2 \right ) \right ) b}{\sqrt {\arccos \left ( \beta \,z \right ) }}} \right ) } \right ) {{\rm e}^{\int ^{y}\!{\frac {s}{b} \left ( \arccos \left ( \gamma \,\RootOf \left ( \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \arccos \left ( \lambda \,{\it \_Z} \right ) {\it \_Z}\,b\lambda \,n-a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }n\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }n{\it \_a}+a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }ny+2\,\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \arccos \left ( \lambda \,{\it \_Z} \right ) {\it \_Z}\,b\lambda +\arccos \left ( \lambda \,{\it \_Z} \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b-\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}bn-2\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-2\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }{\it \_a}+2\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }y- \left ( \arccos \left ( \lambda \,{\it \_Z} \right ) \right ) ^{n+3/2}b\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}-2\,\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b \right ) \right ) \right ) ^{m} \left ( \arccos \left ( \lambda \,\RootOf \left ( \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \arccos \left ( \lambda \,{\it \_Z} \right ) {\it \_Z}\,b\lambda \,n-a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }n\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }n{\it \_a}+a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }ny+2\,\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \arccos \left ( \lambda \,{\it \_Z} \right ) {\it \_Z}\,b\lambda +\arccos \left ( \lambda \,{\it \_Z} \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b-\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}bn-2\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-2\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }{\it \_a}+2\,a\lambda \,\sqrt {\arccos \left ( \lambda \,{\it \_Z} \right ) }y- \left ( \arccos \left ( \lambda \,{\it \_Z} \right ) \right ) ^{n+3/2}b\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}-2\,\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b \right ) \right ) \right ) ^{-n}}{d{\it \_a}}}}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \arccos ^n(\lambda y) w_y + c \arccos ^k(\beta z) w_z = s w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*ArcCos[lambda*y]^n*D[w[x,y,z],y]+c*ArcCos[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*arccos(lambda*y)^n*diff(w(x,y,z),y)+ c*arccos(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 {a{2}^{-n}\sqrt {\pi }}{\lambda \,b} \left ( -{\frac { \left ( \arccos \left ( \lambda \,y \right ) \right ) ^{1-n}{2}^{n}\sqrt {-{\lambda }^{2}{y}^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+{\frac {{2}^{n}\sqrt {\arccos \left ( \lambda \,y \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( \lambda \,y \right ) \right ) \sqrt {-{\lambda }^{2}{y}^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+3\,{\frac {{2}^{-1+n} \left ( -2/3\,n+4/3 \right ) \left ( \lambda \,y\arccos \left ( \lambda \,y \right ) -\sqrt {-{\lambda }^{2}{y}^{2}+1} \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \lambda \,y \right ) \right ) }{\sqrt {\pi } \left ( n-2 \right ) \sqrt {\arccos \left ( \lambda \,y \right ) }}} \right ) }+x,{\frac {{2}^{-n}\sqrt {\pi }}{\lambda } \left ( -{\frac { \left ( \arccos \left ( \lambda \,y \right ) \right ) ^{1-n}{2}^{n}\sqrt {-{\lambda }^{2}{y}^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+{\frac {{2}^{n}\sqrt {\arccos \left ( \lambda \,y \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( \lambda \,y \right ) \right ) \sqrt {-{\lambda }^{2}{y}^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+3\,{\frac {{2}^{-1+n} \left ( -2/3\,n+4/3 \right ) \left ( \lambda \,y\arccos \left ( \lambda \,y \right ) -\sqrt {-{\lambda }^{2}{y}^{2}+1} \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \lambda \,y \right ) \right ) }{\sqrt {\pi } \left ( n-2 \right ) \sqrt {\arccos \left ( \lambda \,y \right ) }}} \right ) }+{\frac {b{2}^{-k} \left ( 2\,{2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\arccos \left ( \beta \,z \right ) \beta \,k-{2}^{k}\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,3/2,\arccos \left ( \beta \,z \right ) \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}-4\,{2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\arccos \left ( \beta \,z \right ) \beta + \left ( \arccos \left ( \beta \,z \right ) \right ) ^{-k+1}{2}^{k}\sqrt {-{\beta }^{2}{z}^{2}+1}\sqrt {\arccos \left ( \beta \,z \right ) }-2\,{2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) k+4\,{2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) \right ) }{c\beta \, \left ( k-2 \right ) \sqrt {\arccos \left ( \beta \,z \right ) }}} \right ) {{\rm e}^{\int \!{\frac {s \left ( \arccos \left ( \lambda \,y \right ) \right ) ^{-n}}{b}}\,{\rm d}y}}\]
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