Added May 31, 2019.
Problem Chapter 6.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arccos ^n(\lambda x) \arccos ^k(\beta z) w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},-\int _1^x\frac {c \cos ^{-1}(\lambda K[1])^n}{a}dK[1]+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta }\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {\sqrt {\pi }\, \left (-\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-n} \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}}{\sqrt {\pi }\, \left (n +2\right )}+\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-n} \arccos \left (\lambda x \right )^{n +1}}{\sqrt {\pi }\, \left (n +2\right )}-\frac {3 \left (\frac {2 n}{3}+\frac {4}{3}\right ) \left (\lambda x \arccos \left (\lambda x \right )-\sqrt {-\lambda ^{2} x^{2}+1}\right ) 2^{-n -1} \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )}{\sqrt {\pi }\, \left (n +2\right ) \sqrt {\arccos \left (\lambda x \right )}}\right ) 2^{n}}{\lambda }+\frac {\left (2 \beta k z 2^{k -1} \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )-4 \beta z 2^{k -1} \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )-2 \sqrt {-\beta ^{2} z^{2}+1}\, k 2^{k -1} \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )-\sqrt {-\beta ^{2} z^{2}+1}\, 2^{k} \LommelS 1 \left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta z \right )\right ) \arccos \left (\beta z \right )+\sqrt {-\beta ^{2} z^{2}+1}\, 2^{k} \arccos \left (\beta z \right )^{-k +1} \sqrt {\arccos \left (\beta z \right )}+4 \sqrt {-\beta ^{2} z^{2}+1}\, 2^{k -1} \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta z \right )\right )\right ) a 2^{-k}}{\left (k -2\right ) \beta c \sqrt {\arccos \left (\beta z \right )}}\right )\]
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Added May 31, 2019.
Problem Chapter 6.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arccos ^n(\lambda x) \arccos ^m(\beta y) \arccos ^k(\gamma z) w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\cos ^{-1}(\gamma z)^{-k} \left (-2 \gamma \cos ^{-1}(\gamma z)^k \int _1^x\frac {c \cos ^{-1}(\lambda K[1])^n \left (\left (\frac {a \cos ^{-1}(\lambda K[1])^{-n} \text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\int _1^x\frac {c \cos ^{-1}(\lambda K[1])^n \cos ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^m}{a}dK[1],\{K[1],1,x\}\right ]}{c}\right ){}^{\frac {1}{m}}\right ){}^m}{a}dK[1]+\left (-i \cos ^{-1}(\gamma z)\right )^k \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\gamma z)\right )+\left (i \cos ^{-1}(\gamma z)\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\gamma z)\right )\right )}{2 \gamma }\right )\right \}\right \}\]
Maple ✓
restart; 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(gamma1*z)^k*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {-\left (k -2\right ) c \gamma 1 \left (\int _{}^{x}\arccos \left (\textit {\_a} \lambda \right )^{n} \arccos \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )^{m}d \textit {\_a} \right )+\frac {\left (\left (k -2\right ) \gamma 1 z \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\gamma 1 z \right )\right ) \arccos \left (\gamma 1 z \right )+\left (-\LommelS 1 \left (-k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\gamma 1 z \right )\right ) \arccos \left (\gamma 1 z \right )+\arccos \left (\gamma 1 z \right )^{-k +\frac {3}{2}}+\left (-k +2\right ) \LommelS 1 \left (-k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\gamma 1 z \right )\right )\right ) \sqrt {-\gamma 1^{2} z^{2}+1}\right ) a 2^{k} 2^{-k}}{\sqrt {\arccos \left (\gamma 1 z \right )}}}{\left (k -2\right ) c \gamma 1}\right )\]
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Added May 31, 2019.
Problem Chapter 6.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arccos ^n(\lambda x) w_y + c \arccos ^k(\beta x) w_z = 0 \]
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*x]^k*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\left (\cos ^{-1}(\beta x)^2\right )^{-k} \left (-c \left (i \cos ^{-1}(\beta x)\right )^k \cos ^{-1}(\beta x)^k \operatorname {Gamma}\left (k+1,-i \cos ^{-1}(\beta x)\right )-c \left (-i \cos ^{-1}(\beta x)\right )^k \cos ^{-1}(\beta x)^k \operatorname {Gamma}\left (k+1,i \cos ^{-1}(\beta x)\right )+2 a \beta z \left (\cos ^{-1}(\beta x)^2\right )^k\right )}{2 a \beta },y-\int _1^x\frac {b \cos ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*arccos(lambda*x)^n*diff(w(x,y,z),y)+c*arccos(beta*x)^k*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (y +\frac {\sqrt {\pi }\, \left (-\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-n} \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}}{\sqrt {\pi }\, \left (n +2\right )}+\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-n} \arccos \left (\lambda x \right )^{n +1}}{\sqrt {\pi }\, \left (n +2\right )}-\frac {3 \left (\frac {2 n}{3}+\frac {4}{3}\right ) \left (\lambda x \arccos \left (\lambda x \right )-\sqrt {-\lambda ^{2} x^{2}+1}\right ) 2^{-n -1} \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )}{\sqrt {\pi }\, \left (n +2\right ) \sqrt {\arccos \left (\lambda x \right )}}\right ) b 2^{n}}{a \lambda }, z +\frac {\sqrt {\pi }\, \left (-\frac {\sqrt {-\beta ^{2} x^{2}+1}\, 2^{-k} \LommelS 1 \left (k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\beta x \right )\right ) \sqrt {\arccos \left (\beta x \right )}}{\sqrt {\pi }\, \left (k +2\right )}+\frac {\sqrt {-\beta ^{2} x^{2}+1}\, 2^{-k} \arccos \left (\beta x \right )^{k +1}}{\sqrt {\pi }\, \left (k +2\right )}-\frac {3 \left (\frac {2 k}{3}+\frac {4}{3}\right ) \left (\beta x \arccos \left (\beta x \right )-\sqrt {-\beta ^{2} x^{2}+1}\right ) 2^{-k -1} \LommelS 1 \left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\beta x \right )\right )}{\sqrt {\pi }\, \left (k +2\right ) \sqrt {\arccos \left (\beta x \right )}}\right ) c 2^{k}}{a \beta }\right )\]
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Added May 31, 2019.
Problem Chapter 6.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arccos ^n(\lambda x) w_y + c \arccos ^k(\beta z) w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta },y-\int _1^x\frac {b \cos ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {\frac {\left (-2 \left (n +2\right ) 2^{-n -1} \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )+\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 +1} \sqrt {\arccos \left (\lambda x \right )}\right ) 2^{-n}\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, b 2^{n}}{\sqrt {\arccos \left (\lambda x \right )}}-\left (n +2\right ) \left (-2 b x 2^{n} 2^{-n -1} \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 (\textit {\_Z} b \lambda n \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )+2 \textit {\_Z} b \lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )-\textit {\_b} a \lambda n \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}+a \lambda n y \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-a \lambda n \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-2 \textit {\_b} a \lambda \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}+2 a \lambda y \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-2 a \lambda \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right ) \sqrt {\arccos \left (\textit {\_Z} \lambda \right )}-\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b n \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right )+\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right ) \arccos \left (\textit {\_Z} \lambda \right )-\sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \arccos \left (\textit {\_Z} \lambda \right )^{n +\frac {3}{2}}-2 \sqrt {-\textit {\_Z}^{2} \lambda ^{2}+1}\, b \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_Z} \lambda \right )\right )\right )\right )^{-n}d \textit {\_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 )\]
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Added May 31, 2019.
Problem Chapter 6.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arccos ^n(\lambda y) w_y + c \arccos ^k(\beta z) w_z = 0 \]
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]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \operatorname {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta },\int _1^y\cos ^{-1}(\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; 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)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \textit {\_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 )\]
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