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]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \cos ^{-1}(\beta x)^n \left (\cos ^{-1}(\beta x)^2\right )^{-n} \left (\left (-i \cos ^{-1}(\beta x)\right )^n \operatorname {Gamma}\left (n+1,i \cos ^{-1}(\beta x)\right )+\left (i \cos ^{-1}(\beta x)\right )^n \operatorname {Gamma}\left (n+1,-i \cos ^{-1}(\beta x)\right )\right )}{2 \beta }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \arccos \left (\beta x \right )^{n}d x}\]
____________________________________________________________________________________
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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {y a_{1} -x a_{2}}{a_{1}}, \frac {z a_{1} -a_{3} x}{a_{1}}\right ) {\mathrm e}^{\frac {-\sqrt {-y^{2} \lambda _{2}^{2}+1}\, a_{1} a_{3} b_{2} \lambda _{1} \lambda _{3} +\left (-\sqrt {-z^{2} \lambda _{3}^{2}+1}\, a_{1} a_{2} b_{3} \lambda _{1} +\left (-\sqrt {-\lambda _{1}^{2} x^{2}+1}\, a_{2} a_{3} b_{1} +\left (a_{2} a_{3} b_{1} x \arccos \left (x \lambda _{1} \right )+\left (a_{2} b_{3} z \arccos \left (z \lambda _{3} \right )+a_{3} b_{2} y \arccos \left (y \lambda _{2} \right )\right ) a_{1} \right ) \lambda _{1} \right ) \lambda _{3} \right ) \lambda _{2}}{a_{1} a_{2} a_{3} \lambda _{1} \lambda _{2} \lambda _{3}}}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.2.3, 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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {\sqrt {\pi }\, \left (-\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-n} \LommelS 1\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}}{\sqrt {\pi }\, \left (n +2\right )}+\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-n} \arccos \left (\lambda x \right )^{n +1}}{\sqrt {\pi }\, \left (n +2\right )}-\frac {3 \left (\frac {2 n}{3}+\frac {4}{3}\right ) \left (\lambda x \arccos \left (\lambda x \right )-\sqrt {-\lambda ^{2} x^{2}+1}\right ) 2^{-n -1} \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )}{\sqrt {\pi }\, \left (n +2\right ) \sqrt {\arccos \left (\lambda x \right )}}\right ) 2^{n}}{\lambda }+\frac {\left (2 \beta k z 2^{k -1} \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )-4 \beta z 2^{k -1} \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )-2 \sqrt {-\beta ^{2} z^{2}+1}\, k 2^{k -1} \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )-\sqrt {-\beta ^{2} z^{2}+1}\, 2^{k} \LommelS 1\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\sqrt {-\beta ^{2} z^{2}+1}\, 2^{k} \arccos \left (\beta z \right )^{-k +1} \sqrt {\arccos \left (\beta z \right )}+4 \sqrt {-\beta ^{2} z^{2}+1}\, 2^{k -1} \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )\right ) a 2^{-k}}{\left (k -2\right ) \beta c \sqrt {\arccos \left (\beta z \right )}}\right ) {\mathrm e}^{\frac {\left (\gamma x \arccos \left (\gamma x \right )-\sqrt {-\gamma ^{2} x^{2}+1}\right ) s}{a \gamma }}\]
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Added Nov 30, 2019.
Problem Chapter 8.7.2.4, 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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {-\left (k -2\right ) c \gamma \left (\int _{}^{x}\arccos \left (\mathit {\_a} \lambda \right )^{n} \arccos \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )^{m}d\mathit {\_a} \right )+\frac {\left (\left (k -2\right ) \gamma z \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\gamma z \right )\right ) \arccos \left (\gamma z \right )+\left (-\LommelS 1\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\gamma z \right )\right ) \arccos \left (\gamma z \right )+\arccos \left (\gamma z \right )^{-k +\frac {3}{2}}+\left (-k +2\right ) \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\gamma z \right )\right )\right ) \sqrt {-\gamma ^{2} z^{2}+1}\right ) a 2^{k} 2^{-k}}{\sqrt {\arccos \left (\gamma z \right )}}}{\left (k -2\right ) c \gamma }\right ) {\mathrm e}^{\frac {s x}{a}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta },\frac {\left (\cos ^{-1}(\lambda x)^2\right )^{-n} \left (-b \left (i \cos ^{-1}(\lambda x)\right )^n \cos ^{-1}(\lambda x)^n \operatorname {Gamma}\left (n+1,-i \cos ^{-1}(\lambda x)\right )-b \left (-i \cos ^{-1}(\lambda x)\right )^n \cos ^{-1}(\lambda x)^n \operatorname {Gamma}\left (n+1,i \cos ^{-1}(\lambda x)\right )+2 a \lambda y \left (\cos ^{-1}(\lambda x)^2\right )^n\right )}{2 a \lambda }\right ) \exp \left (\int _1^z\frac {s \cos ^{-1}\left (\frac {\gamma \left (-a \left (-i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right ) \cos ^{-1}(\beta z)^{-k}-a \left (i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right ) \cos ^{-1}(\beta z)^{-k}+\cos ^{-1}(\beta K[1])^{-k} \left (a \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta K[1])\right ) \left (-i \cos ^{-1}(\beta K[1])\right )^k+2 \beta c x \cos ^{-1}(\beta K[1])^k+a \left (i \cos ^{-1}(\beta K[1])\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta K[1])\right )\right )\right )}{2 \beta c}\right )^m \cos ^{-1}(\beta K[1])^{-k}}{c}dK[1]\right )\right \}\right \}\]
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 ) = \mathit {\_F1} \left (\frac {-\frac {\left (-\LommelS 1\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \arccos \left (\lambda x \right )+\arccos \left (\lambda x \right )^{n +\frac {3}{2}}+\left (n +2\right ) \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, b 2^{n} 2^{-n}}{\sqrt {\arccos \left (\lambda x \right )}}-\left (n +2\right ) \left (-b x 2^{n} 2^{-n} \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}+a y \right ) \lambda }{\left (n +2\right ) a \lambda }, \frac {-\left (k -2\right ) \beta c \left (\int _{}^{y}\arccos \left (\lambda \RootOf \left (\mathit {\_Z} b \lambda n \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )+2 \mathit {\_Z} b \lambda \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )-\mathit {\_b} a \lambda n \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}+a \lambda n y \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-a \lambda n \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-2 \mathit {\_b} a \lambda \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}+2 a \lambda y \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-2 a \lambda \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b n \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right )+\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )-\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \arccos \left (\mathit {\_Z} \lambda \right )^{n +\frac {3}{2}}-2 \sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right )\right )\right )^{-n}d\mathit {\_b} \right )+\frac {\left (\left (k -2\right ) \beta z \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\left (-\LommelS 1\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\arccos \left (\beta z \right )^{-k +\frac {3}{2}}+\left (-k +2\right ) \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )\right ) \sqrt {-\beta ^{2} z^{2}+1}\right ) b 2^{k} 2^{-k}}{\sqrt {\arccos \left (\beta z \right )}}}{\left (k -2\right ) \beta c}\right ) {\mathrm e}^{\int _{}^{y}\frac {s \arccos \left (\gamma \RootOf \left (\mathit {\_Z} b \lambda n \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )+2 \mathit {\_Z} b \lambda \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )-\mathit {\_a} a \lambda n \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}+a \lambda n y \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-a \lambda n \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-2 \mathit {\_a} a \lambda \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}+2 a \lambda y \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-2 a \lambda \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b n \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right )+\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )-\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \arccos \left (\mathit {\_Z} \lambda \right )^{n +\frac {3}{2}}-2 \sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right )\right )\right )^{m} \arccos \left (\lambda \RootOf \left (\mathit {\_Z} b \lambda n \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )+2 \mathit {\_Z} b \lambda \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )-\mathit {\_a} a \lambda n \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}+a \lambda n y \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-a \lambda n \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-2 \mathit {\_a} a \lambda \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}+2 a \lambda y \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-2 a \lambda \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\mathit {\_Z} \lambda \right )}-\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b n \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right )+\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right ) \arccos \left (\mathit {\_Z} \lambda \right )-\sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \arccos \left (\mathit {\_Z} \lambda \right )^{n +\frac {3}{2}}-2 \sqrt {-\mathit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mathit {\_Z} \lambda \right )\right )\right )\right )^{-n}}{b}d\mathit {\_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 ) = \mathit {\_F1} \left (x +\frac {\sqrt {\pi }\, \left (\frac {\sqrt {-\lambda ^{2} y^{2}+1}\, 2^{n} \LommelS 1\left (-n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda y \right )\right ) \sqrt {\arccos \left (\lambda y \right )}}{\sqrt {\pi }\, \left (n -2\right )}-\frac {\sqrt {-\lambda ^{2} y^{2}+1}\, 2^{n} \arccos \left (\lambda y \right )^{-n +1}}{\sqrt {\pi }\, \left (n -2\right )}+\frac {3 \left (-\frac {2 n}{3}+\frac {4}{3}\right ) \left (\lambda y \arccos \left (\lambda y \right )-\sqrt {-\lambda ^{2} y^{2}+1}\right ) 2^{n -1} \LommelS 1\left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda y \right )\right )}{\sqrt {\pi }\, \left (n -2\right ) \sqrt {\arccos \left (\lambda y \right )}}\right ) a 2^{-n}}{b \lambda }, \frac {\sqrt {\pi }\, \left (\frac {\sqrt {-\lambda ^{2} y^{2}+1}\, 2^{n} \LommelS 1\left (-n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda y \right )\right ) \sqrt {\arccos \left (\lambda y \right )}}{\sqrt {\pi }\, \left (n -2\right )}-\frac {\sqrt {-\lambda ^{2} y^{2}+1}\, 2^{n} \arccos \left (\lambda y \right )^{-n +1}}{\sqrt {\pi }\, \left (n -2\right )}+\frac {3 \left (-\frac {2 n}{3}+\frac {4}{3}\right ) \left (\lambda y \arccos \left (\lambda y \right )-\sqrt {-\lambda ^{2} y^{2}+1}\right ) 2^{n -1} \LommelS 1\left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda y \right )\right )}{\sqrt {\pi }\, \left (n -2\right ) \sqrt {\arccos \left (\lambda y \right )}}\right ) 2^{-n}}{\lambda }+\frac {\left (2 \beta k z 2^{k -1} \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )-4 \beta z 2^{k -1} \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )-2 \sqrt {-\beta ^{2} z^{2}+1}\, k 2^{k -1} \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )-\sqrt {-\beta ^{2} z^{2}+1}\, 2^{k} \LommelS 1\left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\sqrt {-\beta ^{2} z^{2}+1}\, 2^{k} \arccos \left (\beta z \right )^{-k +1} \sqrt {\arccos \left (\beta z \right )}+4 \sqrt {-\beta ^{2} z^{2}+1}\, 2^{k -1} \LommelS 1\left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )\right ) b 2^{-k}}{\left (k -2\right ) \beta c \sqrt {\arccos \left (\beta z \right )}}\right ) {\mathrm e}^{\int \frac {s \arccos \left (\lambda y \right )^{-n}}{b}d y}\]
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