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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( \arccos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right ) \exp \left (-\frac {\text {b1} \sqrt {1-\text {lambda1}^2 x^2}}{\text {a1} \text {lambda1}}+\frac {\text {b1} x \cos ^{-1}(\text {lambda1} x)}{\text {a1}}-\frac {\text {b2} \sqrt {1-\text {lambda2}^2 y^2}}{\text {a2} \text {lambda2}}+\frac {\text {b2} y \cos ^{-1}(\text {lambda2} y)}{\text {a2}}-\frac {\text {b3} \sqrt {1-\text {lambda3}^2 z^2}}{\text {a3} \text {lambda3}}+\frac {\text {b3} z \cos ^{-1}(\text {lambda3} z)}{\text {a3}}\right )\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya_{1}-xa_{2}}{a_{1}}},{\frac {za_{1}-xa_{3}}{a_{1}}} \right ) {{\rm e}^{{\frac {1}{a_{1}\,\lambda _{1}\,\lambda _{2}\,a_{2}\,\lambda _{3}\,a_{3}} \left ( -\lambda _{1}\,\lambda _{3}\,\sqrt {-{y}^{2}{\lambda _{2}}^{2}+1}a_{1}\,a_{3}\,b_{2}+ \left ( -\lambda _{1}\,a_{1}\,a_{2}\,b_{3}\,\sqrt {-{z}^{2}{\lambda _{3}}^{2}+1}+ \left ( -a_{2}\,a_{3}\,b_{1}\,\sqrt {-{\lambda _{1}}^{2}{x}^{2}+1}+ \left ( x\arccos \left ( \lambda _{1}\,x \right ) a_{2}\,a_{3}\,b_{1}+a_{1}\, \left ( \arccos \left ( \lambda _{2}\,y \right ) ya_{3}\,b_{2}+\arccos \left ( \lambda _{3}\,z \right ) za_{2}\,b_{3} \right ) \right ) \lambda _{1} \right ) \lambda _{3} \right ) \lambda _{2} \right ) }}}\]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {{2}^{n}\sqrt {\pi }}{\lambda } \left ( {\frac { \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+1}{2}^{-n}}{\sqrt {\pi } \left ( n+2 \right ) }\sqrt {-{\lambda }^{2}{x}^{2}+1}}-{\frac {{2}^{-n}}{\sqrt {\pi } \left ( n+2 \right ) }\sqrt {\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( n+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( \lambda \,x \right ) \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}}-3\,{\frac {{2}^{-1-n} \left ( 2/3\,n+4/3 \right ) \left ( \lambda \,x\arccos \left ( \lambda \,x \right ) -\sqrt {-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt {\pi } \left ( n+2 \right ) \sqrt {\arccos \left ( \lambda \,x \right ) }}} \right ) }+{\frac {a{2}^{-k}}{ \left ( k-2 \right ) \beta \,c} \left ( 2\,\arccos \left ( \beta \,z \right ) {2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\beta \,k-4\,\arccos \left ( \beta \,z \right ) {2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\beta -{2}^{k}\arccos \left ( \beta \,z \right ) \LommelS 1 \left ( -k+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( \beta \,z \right ) \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}-2\,{2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) k+ \left ( \arccos \left ( \beta \,z \right ) \right ) ^{-k+1}{2}^{k}\sqrt {-{\beta }^{2}{z}^{2}+1}\sqrt {\arccos \left ( \beta \,z \right ) }+4\,{2}^{k-1}\sqrt {-{\beta }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) \right ) {\frac {1}{\sqrt {\arccos \left ( \beta \,z \right ) }}}} \right ) {{\rm e}^{{\frac {s}{a\gamma } \left ( \gamma \,x\arccos \left ( \gamma \,x \right ) -\sqrt {-{\gamma }^{2}{x}^{2}+1} \right ) }}}\]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {1}{ \left ( k-2 \right ) \gamma \,c} \left ( -\int ^{x}\! \left ( \arccos \left ( \lambda \,{\it \_a} \right ) \right ) ^{n} \left ( \arccos \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{m}{d{\it \_a}}c\gamma \, \left ( k-2 \right ) +{{2}^{k}{2}^{-k}a \left ( \left ( \left ( -k+2 \right ) \LommelS 1 \left ( -k+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( \gamma \,z \right ) \right ) -\arccos \left ( \gamma \,z \right ) \LommelS 1 \left ( -k+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( \gamma \,z \right ) \right ) + \left ( \arccos \left ( \gamma \,z \right ) \right ) ^{-k+{\frac {3}{2}}} \right ) \sqrt {-{\gamma }^{2}{z}^{2}+1}+\gamma \,\arccos \left ( \gamma \,z \right ) \LommelS 1 \left ( -k+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( \gamma \,z \right ) \right ) z \left ( k-2 \right ) \right ) {\frac {1}{\sqrt {\arccos \left ( \gamma \,z \right ) }}}} \right ) } \right ) {{\rm e}^{{\frac {sx}{a}}}}\]
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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]];
Failed
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 ( -{{2}^{-n}{2}^{n}b \left ( \left ( n+2 \right ) \LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( \lambda \,x \right ) \right ) -\LommelS 1 \left ( n+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( \lambda \,x \right ) \right ) \arccos \left ( \lambda \,x \right ) + \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+{\frac {3}{2}}} \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}{\frac {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+{\frac {1}{2}},{\frac {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+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) \arccos \left ( \lambda \,{\it \_Z} \right ) {\it \_Z}\,b\lambda \,n+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 -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-\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+{\frac {1}{2}},{\frac {1}{2}},\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) bn+\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\arccos \left ( \lambda \,{\it \_Z} \right ) \LommelS 1 \left ( n+{\frac {3}{2}},{\frac {3}{2}},\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) b-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-\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}b \left ( \arccos \left ( \lambda \,{\it \_Z} \right ) \right ) ^{n+{\frac {3}{2}}}-2\,\sqrt {-{{\it \_Z}}^{2}{\lambda }^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,{\it \_Z} \right ) \right ) b \right ) \right ) \right ) ^{-n}{d{\it \_b}}c\beta \, \left ( k-2 \right ) +{{2}^{-k}{2}^{k}b \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 ) {{\rm e}^{\int ^{y}\!{\frac {s}{b} \left ( \arccos \left ( \gamma \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b \left ( \arccos \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}}{a}}{d{\it \_a}}-y+\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x \right ) \right ) \right ) ^{m} \left ( \arccos \left ( \lambda \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b \left ( \arccos \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}}{a}}{d{\it \_a}}-y+\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x \right ) \right ) \right ) ^{-n}}{d{\it \_b}}}}\]
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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 ) ^{-n+1}{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 ) ^{-n+1}{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 ) {2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\beta \,k-4\,\arccos \left ( \beta \,z \right ) {2}^{k-1}\LommelS 1 \left ( -k+1/2,1/2,\arccos \left ( \beta \,z \right ) \right ) z\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 ) {{\rm e}^{\int \!{\frac {s \left ( \arccos \left ( \lambda \,y \right ) \right ) ^{-n}}{b}}\,{\rm d}y}}\]
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