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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},-\int _1^x\frac {c \cos ^{-1}(\lambda K[1])^n}{a}dK[1]+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta }\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {{2}^{n}\sqrt {\pi }}{\lambda } \left ( {\frac { \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+1}{2}^{-n}\sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( n+2 \right ) }}-{\frac {{2}^{-n}\sqrt {\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( n+2 \right ) }}-3\,{\frac {{2}^{-n-1} \left ( 2/3\,n+4/3 \right ) \left ( x\lambda \,\arccos \left ( \lambda \,x \right ) -\sqrt {-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt {\pi } \left ( n+2 \right ) \sqrt {\arccos \left ( \lambda \,x \right ) }}} \right ) }+{\frac {a{2}^{-k} \left ( 2\,\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\arccos \left ( \beta \,z \right ) {2}^{k-1}\beta \,k-{2}^{k}\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,3/2,\arccos \left ( \beta \,z \right ) \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}-4\,\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\arccos \left ( \beta \,z \right ) {2}^{k-1}\beta + \left ( \arccos \left ( \beta \,z \right ) \right ) ^{1-k}{2}^{k}\sqrt {-{\beta }^{2}{z}^{2}+1}\sqrt {\arccos \left ( \beta \,z \right ) }-2\,\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) {2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}k+4\,\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) {2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1} \right ) }{ \left ( k-2 \right ) \beta \,c\sqrt {\arccos \left ( \beta \,z \right ) }}} \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\cos ^{-1}(\gamma z)^{-k} \left (-2 \gamma \cos ^{-1}(\gamma z)^k \int _1^x\frac {c \cos ^{-1}(\lambda K[1])^n \left (\left (\frac {a \cos ^{-1}(\lambda K[1])^{-n} \text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\int _1^x\frac {c \cos ^{-1}(\lambda K[1])^n \cos ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^m}{a}dK[1],\{K[1],1,x\}\right ]}{c}\right ){}^{\frac {1}{m}}\right ){}^m}{a}dK[1]+\left (-i \cos ^{-1}(\gamma z)\right )^k \text {Gamma}\left (1-k,-i \cos ^{-1}(\gamma z)\right )+\left (i \cos ^{-1}(\gamma z)\right )^k \text {Gamma}\left (1-k,i \cos ^{-1}(\gamma z)\right )\right )}{2 \gamma }\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {1}{ \left ( k-2 \right ) \gamma 1\,c} \left ( -\int ^{x}\! \left ( \arccos \left ( {\it \_a}\,\lambda \right ) \right ) ^{n} \left ( \arccos \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta }{a}} \right ) \right ) ^{m}{d{\it \_a}}c\gamma 1\, \left ( k-2 \right ) +{\frac {{2}^{-k}{2}^{k} \left ( \left ( \left ( 2-k \right ) \LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \gamma 1\,z \right ) \right ) -\arccos \left ( \gamma 1\,z \right ) \LommelS 1 \left ( -k+3/2,3/2,\arccos \left ( \gamma 1\,z \right ) \right ) + \left ( \arccos \left ( \gamma 1\,z \right ) \right ) ^{-k+3/2} \right ) \sqrt {-{\gamma 1}^{2}{z}^{2}+1}+\gamma 1\,\arccos \left ( \gamma 1\,z \right ) \LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \gamma 1\,z \right ) \right ) z \left ( k-2 \right ) \right ) a}{\sqrt {\arccos \left ( \gamma 1\,z \right ) }}} \right ) } \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\left (\cos ^{-1}(\beta x)^2\right )^{-k} \left (-c \left (i \cos ^{-1}(\beta x)\right )^k \cos ^{-1}(\beta x)^k \text {Gamma}\left (k+1,-i \cos ^{-1}(\beta x)\right )-c \left (-i \cos ^{-1}(\beta x)\right )^k \cos ^{-1}(\beta x)^k \text {Gamma}\left (k+1,i \cos ^{-1}(\beta x)\right )+2 a \beta z \left (\cos ^{-1}(\beta x)^2\right )^k\right )}{2 a \beta },y-\int _1^x\frac {b \cos ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {b{2}^{n}\sqrt {\pi }}{a\lambda } \left ( {\frac { \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+1}{2}^{-n}\sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( n+2 \right ) }}-{\frac {{2}^{-n}\sqrt {\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( n+2 \right ) }}-3\,{\frac {{2}^{-n-1} \left ( 2/3\,n+4/3 \right ) \left ( x\lambda \,\arccos \left ( \lambda \,x \right ) -\sqrt {-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt {\pi } \left ( n+2 \right ) \sqrt {\arccos \left ( \lambda \,x \right ) }}} \right ) }+y,{\frac {c{2}^{k}\sqrt {\pi }}{a\beta } \left ( {\frac { \left ( \arccos \left ( \beta \,x \right ) \right ) ^{k+1}{2}^{-k}\sqrt {-{\beta }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( 2+k \right ) }}-{\frac {{2}^{-k}\sqrt {\arccos \left ( \beta \,x \right ) }\LommelS 1 \left ( 3/2+k,3/2,\arccos \left ( \beta \,x \right ) \right ) \sqrt {-{\beta }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( 2+k \right ) }}-3\,{\frac {{2}^{-1-k} \left ( 4/3+2/3\,k \right ) \left ( \beta \,x\arccos \left ( \beta \,x \right ) -\sqrt {-{\beta }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \beta \,x \right ) \right ) }{\sqrt {\pi } \left ( 2+k \right ) \sqrt {\arccos \left ( \beta \,x \right ) }}} \right ) }+z \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta },y-\int _1^x\frac {b \cos ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right \}\right \}\]
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 ) ={\it \_F1} \left ( {\frac {1}{ \left ( n+2 \right ) a\lambda } \left ( {\frac {b \left ( -2\,{2}^{-n-1} \left ( n+2 \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) +{2}^{-n} \left ( \arccos \left ( \lambda \,x \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) -\sqrt {\arccos \left ( \lambda \,x \right ) } \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+1} \right ) \right ) {2}^{n}\sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\arccos \left ( \lambda \,x \right ) }}}-\lambda \, \left ( n+2 \right ) \left ( -2\,\sqrt {\arccos \left ( \lambda \,x \right ) }bx{2}^{n}{2}^{-n-1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) +ya \right ) \right ) },{\frac {1}{ \left ( k-2 \right ) \beta \,c} \left ( -\int ^{y}\! \left ( \arccos \left ( \lambda \,\RootOf \left ( \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) \arccos \left ( {\it \_Z}\,\lambda \right ) {\it \_Z}\,b\lambda \,n+2\,\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) \arccos \left ( {\it \_Z}\,\lambda \right ) {\it \_Z}\,b\lambda -a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }n\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-{\it \_b}\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }n+a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }ny-\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) bn+\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) \arccos \left ( {\it \_Z}\,\lambda \right ) b-2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-2\,{\it \_b}\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }+2\,a\lambda \,\sqrt {\arccos \left ( {\it \_Z}\,\lambda \right ) }y-2\,\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_Z}\,\lambda \right ) \right ) b- \left ( \arccos \left ( {\it \_Z}\,\lambda \right ) \right ) ^{n+3/2}\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b \right ) \right ) \right ) ^{-n}{d{\it \_b}}c\beta \, \left ( k-2 \right ) +{\frac {b \left ( \left ( \left ( 2-k \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 ) {2}^{k}{2}^{-k}}{\sqrt {\arccos \left ( \beta \,z \right ) }}} \right ) } \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x}{a}+\frac {\cos ^{-1}(\beta z)^{-k} \left (\left (-i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,-i \cos ^{-1}(\beta z)\right )+\left (i \cos ^{-1}(\beta z)\right )^k \text {Gamma}\left (1-k,i \cos ^{-1}(\beta z)\right )\right )}{2 \beta },\int _1^y\cos ^{-1}(\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right )\right \}\right \}\]
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 ) ={\it \_F1} \left ( {\frac {a{2}^{-n}\sqrt {\pi }}{\lambda \,b} \left ( -{\frac { \left ( \arccos \left ( y\lambda \right ) \right ) ^{-n+1}{2}^{n}\sqrt {-{y}^{2}{\lambda }^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+{\frac {{2}^{n}\sqrt {\arccos \left ( y\lambda \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( y\lambda \right ) \right ) \sqrt {-{y}^{2}{\lambda }^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+3\,{\frac {{2}^{n-1} \left ( -2/3\,n+4/3 \right ) \left ( y\lambda \,\arccos \left ( y\lambda \right ) -\sqrt {-{y}^{2}{\lambda }^{2}+1} \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( y\lambda \right ) \right ) }{\sqrt {\pi } \left ( n-2 \right ) \sqrt {\arccos \left ( y\lambda \right ) }}} \right ) }+x,{\frac {{2}^{-n}\sqrt {\pi }}{\lambda } \left ( -{\frac { \left ( \arccos \left ( y\lambda \right ) \right ) ^{-n+1}{2}^{n}\sqrt {-{y}^{2}{\lambda }^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+{\frac {{2}^{n}\sqrt {\arccos \left ( y\lambda \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( y\lambda \right ) \right ) \sqrt {-{y}^{2}{\lambda }^{2}+1}}{\sqrt {\pi } \left ( n-2 \right ) }}+3\,{\frac {{2}^{n-1} \left ( -2/3\,n+4/3 \right ) \left ( y\lambda \,\arccos \left ( y\lambda \right ) -\sqrt {-{y}^{2}{\lambda }^{2}+1} \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( y\lambda \right ) \right ) }{\sqrt {\pi } \left ( n-2 \right ) \sqrt {\arccos \left ( y\lambda \right ) }}} \right ) }-{\frac {b{2}^{-k} \left ( -2\,\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\arccos \left ( \beta \,z \right ) {2}^{k-1}\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\,\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\arccos \left ( \beta \,z \right ) {2}^{k-1}\beta +2\,\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) {2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}k- \left ( \arccos \left ( \beta \,z \right ) \right ) ^{1-k}{2}^{k}\sqrt {-{\beta }^{2}{z}^{2}+1}\sqrt {\arccos \left ( \beta \,z \right ) }-4\,\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) {2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1} \right ) }{ \left ( k-2 \right ) \beta \,c\sqrt {\arccos \left ( \beta \,z \right ) }}} \right ) \]
____________________________________________________________________________________