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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {a k y \arccos \left (\frac {y}{\beta }\right )+b c x \arccos \left (\frac {x}{\lambda }\right )-\sqrt {\frac {\beta ^{2}-y^{2}}{\beta ^{2}}}\, a \beta k -\sqrt {-\frac {x^{2}}{\lambda ^{2}}+1}\, b c \lambda }{a b}}\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (\left (\beta y +\lambda x \right ) \arccos \left (\beta y +\lambda x \right )-\sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -\lambda ^{2} x^{2}+1}\right ) c}{a \lambda +b \beta }}\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (\frac {a \arcsin \left (\beta y +\lambda x \right )}{2}+\left (2 b \beta x +\left (-\beta y +\lambda x \right ) a \right ) \left (\beta y +\lambda x \right ) \arccos \left (\beta y +\lambda x \right )+\left (-2 b \beta x +\left (\frac {3 \beta y}{2}-\frac {\lambda x}{2}\right ) a \right ) \sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -\lambda ^{2} x^{2}+1}\right ) a}{2 \left (a \lambda +b \beta \right )^{2}}}\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}+\arccos \left (\lambda x \right )^{n +1}+\frac {\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )}{\sqrt {\arccos \left (\lambda x \right )}}\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, b 2^{n} 2^{-n}+\left (n +2\right ) \left (-b x 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}+a y \right ) \lambda }{\left (n +2\right ) a \lambda }\right ) {\mathrm e}^{\int _{}^{x}\frac {c \arccos \left (\textit {\_b} \mu \right )^{m}+s \left (-\arccos \left (\frac {\left (\left (-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_b} \lambda \right )\right ) \sqrt {\arccos \left (\textit {\_b} \lambda \right )}+\arccos \left (\textit {\_b} \lambda \right )^{n +1}+\frac {\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_b} \lambda \right )\right )}{\sqrt {\arccos \left (\textit {\_b} \lambda \right )}}\right ) \sqrt {-\textit {\_b}^{2} \lambda ^{2}+1}\, b 2^{n} 2^{-n}+\left (n +2\right ) \left (-\textit {\_b} b 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_b} \lambda \right )\right ) \sqrt {\arccos \left (\textit {\_b} \lambda \right )}-a y +a \left (\int \frac {b \arccos \left (\lambda x \right )^{n}}{a}d x \right )\right ) \lambda \right ) \beta }{\left (n +2\right ) a \lambda }\right )+\pi \right )^{k}}{a}d \textit {\_b}}\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (\LommelS 1 \left (-n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda y \right )\right ) \sqrt {\arccos \left (\lambda y \right )}-\arccos \left (\lambda y \right )^{-n +1}+\frac {\left (n -2\right ) \LommelS 1 \left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda y \right )\right )}{\sqrt {\arccos \left (\lambda y \right )}}\right ) \sqrt {-\lambda ^{2} y^{2}+1}\, a 2^{n} 2^{-n}-\left (n -2\right ) \left (a y 2^{n} 2^{-n} \LommelS 1 \left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda y \right )\right ) \sqrt {\arccos \left (\lambda y \right )}-b x \right ) \lambda }{\left (n -2\right ) b \lambda }\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (c \arccos \left (\frac {\left (-\left (\LommelS 1 \left (-n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_b} \lambda \right )\right ) \sqrt {\arccos \left (\textit {\_b} \lambda \right )}-\arccos \left (\textit {\_b} \lambda \right )^{-n +1}+\frac {\left (n -2\right ) \LommelS 1 \left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_b} \lambda \right )\right )}{\sqrt {\arccos \left (\textit {\_b} \lambda \right )}}\right ) \sqrt {-\textit {\_b}^{2} \lambda ^{2}+1}\, a 2^{n} 2^{-n}+\left (n -2\right ) \left (\textit {\_b} a 2^{n} 2^{-n} \LommelS 1 \left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_b} \lambda \right )\right ) \sqrt {\arccos \left (\textit {\_b} \lambda \right )}-a \left (\int \arccos \left (\lambda y \right )^{-n}d y \right )+b x \right ) \lambda \right ) \mu }{\left (n -2\right ) b \lambda }\right )^{m}+s \arccos \left (\textit {\_b} \beta \right )^{k}\right ) \arccos \left (\textit {\_b} \lambda \right )^{-n}}{b}d \textit {\_b}}\]
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