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