Added June 26, 2019.
Problem Chapter 7.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \arcsin ^k(\lambda x)+s \]
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*ArcSin[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {\left (\sin ^{-1}(\lambda x)^2\right )^{-k} \left (-i c \left (i \sin ^{-1}(\lambda x)\right )^k \sin ^{-1}(\lambda x)^k \text {Gamma}\left (k+1,-i \sin ^{-1}(\lambda x)\right )+i c \left (-i \sin ^{-1}(\lambda x)\right )^k \sin ^{-1}(\lambda x)^k \text {Gamma}\left (k+1,i \sin ^{-1}(\lambda x)\right )+2 \lambda s x \left (\sin ^{-1}(\lambda x)^2\right )^k\right )}{2 \lambda }\right \}\right \}\]
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*arcsin(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \arcsin \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]
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Added June 26, 2019.
Problem Chapter 7.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 w_x + a_2 w_y + a_3 w_z = b_1 \arcsin (\lambda _1 x)+b_2 \arcsin (\lambda _2 y)+b_3 \arcsin (\lambda _3 z) \]
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*ArcSin[lambda1*x]+b2*ArcSin[lambda2*y]+b3*ArcSin[lambda3*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right )+\frac {\text {b1} \sqrt {1-\text {lambda1}^2 x^2}}{\text {a1} \text {lambda1}}+\frac {\text {b1} x \sin ^{-1}(\text {lambda1} x)}{\text {a1}}+\frac {\text {b2} \sqrt {1-\text {lambda2}^2 y^2}}{\text {a2} \text {lambda2}}+\frac {\text {b2} y \sin ^{-1}(\text {lambda2} y)}{\text {a2}}+\frac {\text {b3} \sqrt {1-\text {lambda3}^2 z^2}}{\text {a3} \text {lambda3}}+\frac {\text {b3} z \sin ^{-1}(\text {lambda3} z)}{\text {a3}}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a1*diff(w(x,y,z),x)+ a2*diff(w(x,y,z),y)+ a3*diff(w(x,y,z),z)= b1*arcsin(lambda1*x)+b2*arcsin(lambda2*y)+b3*arcsin(lambda3*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{\lambda 1\,{\it a1}\,{\it a2}\,\lambda 2\,{\it a3}\,\lambda 3} \left ( {\it \_F1} \left ( {\frac {y{\it a1}-{\it a2}\,x}{{\it a1}}},{\frac {z{\it a1}-{\it a3}\,x}{{\it a1}}} \right ) {\it a1}\,\lambda 1\,{\it a2}\,\lambda 2\,{\it a3}\,\lambda 3+\sqrt {-{\lambda 1}^{2}{x}^{2}+1}{\it a2}\,{\it a3}\,{\it b1}\,\lambda 2\,\lambda 3+ \left ( \lambda 3\,{\it a1}\,{\it a3}\,{\it b2}\,\sqrt {-{\lambda 2}^{2}{y}^{2}+1}+ \left ( x\lambda 3\,\arcsin \left ( \lambda 1\,x \right ) {\it a2}\,{\it a3}\,{\it b1}+{\it a1}\, \left ( \lambda 3\,\arcsin \left ( \lambda 2\,y \right ) y{\it a3}\,{\it b2}+{\it b3}\, \left ( \lambda 3\,\arcsin \left ( \lambda 3\,z \right ) z+\sqrt {-{\lambda 3}^{2}{z}^{2}+1} \right ) {\it a2} \right ) \right ) \lambda 2 \right ) \lambda 1 \right ) }\]
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Added June 26, 2019.
Problem Chapter 7.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arcsin ^n(\lambda x) \arcsin ^k(\beta z) w_z = s \arcsin ^m(\gamma x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*ArcSin[lambda*x]^n*ArcSin[beta*z]^k*D[w[x,y,z],z]== s*ArcSin[gamma*x]^m; 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*arcsin(lambda*x)^n*arcsin(beta*z)^k*diff(w(x,y,z),z)= s*arcsin(gamma*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!{\frac {s \left ( \arcsin \left ( x\gamma \right ) \right ) ^{m}}{a}}\,{\rm d}x+{\it \_F1} \left ( {\frac {ay-xb}{a}},-2\,{\frac {1}{ \left ( 1+n \right ) \lambda \, \left ( {x}^{2}{\lambda }^{2}-1 \right ) c\beta \, \left ( k-1 \right ) } \left ( 1/2\,{2}^{n}\lambda \,{2}^{-n}cx\beta \, \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x\lambda \right ) \right ) }{\sqrt {\arcsin \left ( x\lambda \right ) }}}+ \left ( \arcsin \left ( x\lambda \right ) \right ) ^{n} \right ) \left ( -{x}^{2}{\lambda }^{2}+1 \right ) +1/2\,{2}^{n}{2}^{-n}\arcsin \left ( x\lambda \right ) c\beta \, \left ( x\lambda -1 \right ) \left ( x\lambda +1 \right ) \left ( k-1 \right ) \left ( -{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x\lambda \right ) \right ) }{\sqrt {\arcsin \left ( x\lambda \right ) }}}+ \left ( \arcsin \left ( x\lambda \right ) \right ) ^{n} \right ) \sqrt {-{x}^{2}{\lambda }^{2}+1}+ \left ( x\lambda -1 \right ) \left ( x\lambda +1 \right ) \left ( -1/2\,{\frac {a{2}^{-k}{2}^{k} \left ( 1+n \right ) \left ( \arcsin \left ( \beta \,z \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) - \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k} \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{3/2} \right ) \sqrt {-{\beta }^{2}{z}^{2}+1}}{\sqrt {\arcsin \left ( \beta \,z \right ) }}}+ \left ( 1/2\,{\frac {a{2}^{-k}z{2}^{k} \left ( 1+n \right ) \LommelS 1 \left ( -k+3/2,1/2,\arcsin \left ( \beta \,z \right ) \right ) }{\sqrt {\arcsin \left ( \beta \,z \right ) }}}-1/2\,a{2}^{-k}\sqrt {\arcsin \left ( \beta \,z \right ) }kz{2}^{k} \left ( 1+n \right ) \LommelS 1 \left ( -k+1/2,3/2,\arcsin \left ( \beta \,z \right ) \right ) +1/2\,{2}^{n}{2}^{-n}c\sqrt {\arcsin \left ( x\lambda \right ) }nx \left ( k-1 \right ) \LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x\lambda \right ) \right ) +z \left ( \arcsin \left ( \beta \,z \right ) \right ) ^{-k} \left ( 1+n \right ) \left ( {2}^{k-1}-1/2\,{2}^{k} \right ) a{2}^{-k}+{2}^{n}{2}^{-n-1}c \left ( \arcsin \left ( x\lambda \right ) \right ) ^{n}x \left ( k-1 \right ) \right ) \beta \right ) \lambda \right ) } \right ) \]
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Added June 26, 2019.
Problem Chapter 7.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arcsin ^n(\lambda x) \arcsin ^m(\beta y) \arcsin ^k(\gamma z) w_z = s \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*ArcSin[lambda*x]^n*ArcSin[beta*y]^m*ArcSin[gamma*z]^k*D[w[x,y,z],z]== s; 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*arcsin(lambda*x)^n*arcsin(beta*y)^m*arcsin(gamma*z)^k*diff(w(x,y,z),z)= s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {sx}{a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}},-\int ^{x}\! \left ( \arcsin \left ( {\it \_a}\,\lambda \right ) \right ) ^{n} \left ( \arcsin \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{m}{d{\it \_a}}-{\frac {a{2}^{-k}}{ \left ( k-1 \right ) c\gamma } \left ( -{2}^{k}\arcsin \left ( \gamma \,z \right ) k\LommelS 1 \left ( -k+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \gamma \,z \right ) \right ) z\gamma +{2}^{k}\sqrt {-{\gamma }^{2}{z}^{2}+1} \left ( \arcsin \left ( \gamma \,z \right ) \right ) ^{-k+{\frac {3}{2}}}+z\LommelS 1 \left ( -k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \gamma \,z \right ) \right ) {2}^{k}\gamma -\sqrt {-{\gamma }^{2}{z}^{2}+1}\LommelS 1 \left ( -k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \gamma \,z \right ) \right ) \arcsin \left ( \gamma \,z \right ) {2}^{k} \right ) {\frac {1}{\sqrt {\arcsin \left ( \gamma \,z \right ) }}}} \right ) \]
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Added June 26, 2019.
Problem Chapter 7.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arcsin ^n(\lambda x) w_y + c \arcsin ^k(\beta z) w_z = s \arcsin ^m(\gamma x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*ArcSin[lambda*x]^n*D[w[x, y,z], y] + c*ArcSin[beta*z]^k*D[w[x,y,z],z]== s*ArcSin[gamma*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed Generates Solve::incnst: Inconsistent or redundant transcendental equation
Maple ✗
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*arcsin(lambda*x)^n*diff(w(x,y,z),y)+ c*arcsin(beta*z)^k*diff(w(x,y,z),z)= s*arcsin(gamma*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
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Added June 26, 2019.
Problem Chapter 7.7.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arcsin ^n(\lambda x) w_y + c \arcsin ^k(\beta z) w_z = s \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*ArcSin[lambda*x]^n*D[w[x, y,z], y] + c*ArcSin[beta*z]^k*D[w[x,y,z],z]== s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed Generates Solve::incnst: Inconsistent or redundant transcendental equation
Maple ✗
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*arcsin(lambda*x)^n*diff(w(x,y,z),y)+ c*arcsin(beta*z)^k*diff(w(x,y,z),z)= s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
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