Added March 9, 2019.
Problem Chapter 4.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c \arccos (\frac {x}{\lambda } + k \arccos (\frac {y}{\beta } ) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*ArcCos[x/lambda] + k*ArcCos[y/beta])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {-\frac {k \left (\sqrt {a^2 \left (\beta ^2-y^2\right )} (a y-b x) \tan ^{-1}\left (\frac {a y}{\sqrt {a^2 \left (\beta ^2-y^2\right )}}\right )+a^2 \left (\beta ^2-y^2\right )\right )}{b \beta \sqrt {1-\frac {y^2}{\beta ^2}}}+a k x \cos ^{-1}\left (\frac {y}{\beta }\right )-a c \lambda \sqrt {1-\frac {x^2}{\lambda ^2}}+a c x \cos ^{-1}\left (\frac {x}{\lambda }\right )}{a^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*arccos(x/lambda)+k*arccos(y/beta))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {1}{ab} \left ( a\arccos \left ( {\frac {y}{\beta }} \right ) ky-\sqrt {{\frac {{\beta }^{2}-{y}^{2}}{{\beta }^{2}}}}a\beta \,k-\sqrt {-{\frac {{x}^{2}}{{\lambda }^{2}}}+1}bc\lambda +\arccos \left ( {\frac {x}{\lambda }} \right ) bcx \right ) }}}\]
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Added March 9, 2019.
Problem Chapter 4.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arccos (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcCos[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c \left (\beta (b x-a y) \sin ^{-1}(\beta y+\lambda x)+x (a \lambda +b \beta ) \cos ^{-1}(\beta y+\lambda x)+a \left (-\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}\right )\right )}{a (a \lambda +b \beta )}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*arccos(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={{\rm e}^{{\frac { \left ( -\sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}+\arccos \left ( \beta \,y+\lambda \,x \right ) \left ( \beta \,y+\lambda \,x \right ) \right ) c}{a\lambda +b\beta }}}}{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]
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Added March 9, 2019.
Problem Chapter 4.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = a x \arccos (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == a*x*ArcCos[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {\left (a^2+2 \beta ^2 (b x-a y)^2\right ) \sin ^{-1}(\beta y+\lambda x)-a \sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1} (-3 a \beta y+a \lambda x+4 b \beta x)+2 x^2 (a \lambda +b \beta )^2 \cos ^{-1}(\beta y+\lambda x)}{4 (a \lambda +b \beta )^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arccos(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{1/2\,{\frac { \left ( \left ( \left ( -1/2\,\lambda \,x+3/2\,\beta \,y \right ) a-2\,bx\beta \right ) \sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}+ \left ( \beta \,y+\lambda \,x \right ) \left ( \left ( -\beta \,y+\lambda \,x \right ) a+2\,bx\beta \right ) \arccos \left ( \beta \,y+\lambda \,x \right ) +1/2\,\arcsin \left ( \beta \,y+\lambda \,x \right ) a \right ) a}{ \left ( a\lambda +b\beta \right ) ^{2}}}}}\]
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Added March 9, 2019.
Problem Chapter 4.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccos ^n(\lambda x)w_y = \left ( c \arccos ^m(\mu x) + s \arccos ^k(\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCos[lambda*x]^n*D[w[x, y], y] == (c*ArcCos[mu*x]^m + s*ArcCos[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac {b \cos ^{-1}(\lambda K[1])^n}{a}dK[1]\right ) \exp \left (\int _1^x\frac {s \cos ^{-1}\left (\beta \left (y-\int _1^x\frac {b \cos ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac {b \cos ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k+c \cos ^{-1}(\mu K[2])^m}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arccos(lambda*x)^n*diff(w(x,y),y) =(c*arccos(mu*x)^m+s*arccos(beta*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( n+2 \right ) a\lambda } \left ( b{2}^{-n}{2}^{n} \left ( {\frac { \left ( n+2 \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt {\arccos \left ( \lambda \,x \right ) }}}-\sqrt {\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) + \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+1} \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}+\lambda \, \left ( n+2 \right ) \left ( -\sqrt {\arccos \left ( \lambda \,x \right ) }bx{2}^{n}{2}^{-n}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) +ya \right ) \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( c \left ( \arccos \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( \pi -\arccos \left ( {\frac {\beta }{ \left ( n+2 \right ) a\lambda } \left ( b{2}^{n} \left ( {\frac { \left ( n+2 \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) }{\sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }}}-\sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) + \left ( \arccos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n+1} \right ) {2}^{-n}\sqrt {-{{\it \_b}}^{2}{\lambda }^{2}+1}+\lambda \, \left ( n+2 \right ) \left ( -{2}^{-n}\sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }b\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) {\it \_b}\,{2}^{n}+a\int \!{\frac {b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-ya \right ) \right ) } \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]
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Added March 9, 2019.
Problem Chapter 4.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccos ^n(\lambda y)w_y = \left ( c \arccos ^m(\mu x) + s \arccos ^k(\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCos[lambda*y]^n*D[w[x, y], y] == (c*ArcCos[mu*x]^m + s*ArcCos[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\cos ^{-1}(\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right ) \exp \left (\int _1^y\frac {\cos ^{-1}(\lambda K[2])^{-n} \left (s \cos ^{-1}(\beta K[2])^k+c \cos ^{-1}\left (\frac {\mu \left (b x-a \int _1^y\cos ^{-1}(\lambda K[1])^{-n}dK[1]+a \int _1^{K[2]}\cos ^{-1}(\lambda K[1])^{-n}dK[1]\right )}{b}\right ){}^m\right )}{b}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arccos(lambda*y)^n*diff(w(x,y),y) =(c*arccos(mu*x)^m+s*arccos(beta*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( n-2 \right ) \lambda \,b} \left ( {2}^{-n}{2}^{n} \left ( {\frac { \left ( n-2 \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( y\lambda \right ) \right ) }{\sqrt {\arccos \left ( y\lambda \right ) }}}+\sqrt {\arccos \left ( y\lambda \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( y\lambda \right ) \right ) - \left ( \arccos \left ( y\lambda \right ) \right ) ^{-n+1} \right ) a\sqrt {-{y}^{2}{\lambda }^{2}+1}-\lambda \, \left ( n-2 \right ) \left ( a\sqrt {\arccos \left ( y\lambda \right ) }\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( y\lambda \right ) \right ) y{2}^{n}{2}^{-n}-bx \right ) \right ) } \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \arccos \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \arccos \left ( {\frac {\mu }{ \left ( n-2 \right ) \lambda \,b} \left ( -{2}^{n} \left ( {\frac { \left ( n-2 \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) }{\sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }}}+\sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) - \left ( \arccos \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n+1} \right ) a{2}^{-n}\sqrt {-{{\it \_b}}^{2}{\lambda }^{2}+1}+\lambda \, \left ( n-2 \right ) \left ( a{2}^{-n}\sqrt {\arccos \left ( {\it \_b}\,\lambda \right ) }\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( {\it \_b}\,\lambda \right ) \right ) {\it \_b}\,{2}^{n}-a\int \! \left ( \arccos \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y+bx \right ) \right ) } \right ) \right ) ^{m}+s \left ( \arccos \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]
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