Added April 13, 2019.
Problem Chapter 5.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \arcsin ^k(\lambda x) + c_2 \arcsin ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcSin[lambda*x]^k+c2*ArcSin[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \sin ^{-1}(\lambda K[1])^k+\text {c2} \sin ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*arcsin(lambda*x)^k+c2*arcsin(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \arcsin \left (\mathit {\_a} \lambda \right )^{k}+\mathit {c2} \arcsin \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n}\right ) {\mathrm e}^{-\frac {\mathit {\_a}}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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Added April 13, 2019.
Problem Chapter 5.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \arcsin ^k(\lambda x) \arcsin ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcSin[lambda*x]^k*ArcSin[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \sin ^{-1}(\lambda K[1])^k \sin ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ arcsin(lambda*x)^k*arcsin(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\arcsin \left (\mathit {\_a} \lambda \right )^{k} \arcsin \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {\mathit {\_a} c}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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Added April 13, 2019.
Problem Chapter 5.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c_1 \arcsin (\lambda _1 x) + c_2 \arcsin (\lambda _2 y)\right ) w+ s_1 \arcsin ^n(\beta _1 x)+ s_2 \arcsin ^k(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcSin[lambda1*x] + c2*ArcSin[lambda2*y])*w[x,y]+ s1*ArcSin[beta1*x]^n+ s2*ArcSin[beta2*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} \sqrt {1-\text {lambda1}^2 x^2}}{a \text {lambda1}}+\frac {\text {c1} x \sin ^{-1}(\text {lambda1} x)}{a}+\frac {\text {c2} \sqrt {1-\text {lambda2}^2 y^2}}{b \text {lambda2}}+\frac {\text {c2} y \sin ^{-1}(\text {lambda2} y)}{b}\right ) \left (\int _1^x\frac {\exp \left (-\frac {b \text {c1} \text {lambda2} \sin ^{-1}(\text {lambda1} K[1]) K[1] \text {lambda1}+\text {c2} \text {lambda2} \sin ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) (a y+b (K[1]-x)) \text {lambda1}+a \text {c2} \sqrt {-y^2 \text {lambda2}^2-\frac {b^2 (x-K[1])^2 \text {lambda2}^2}{a^2}+\frac {2 b y (x-K[1]) \text {lambda2}^2}{a}+1} \text {lambda1}+b \text {c1} \text {lambda2} \sqrt {1-\text {lambda1}^2 K[1]^2}}{a b \text {lambda1} \text {lambda2}}\right ) \left (\text {s2} \sin ^{-1}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )^k+\text {s1} \sin ^{-1}(\text {beta1} K[1])^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arcsin(lambda1*x) + c2*arcsin(lambda2*y))*w(x,y)+ s1*arcsin(beta1*x)^n+ s2*arcsin(beta2*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {s1} \arcsin \left (\mathit {\_a} \beta 1 \right )^{n}+\mathit {s2} \arcsin \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta 2}{a}\right )^{k}\right ) {\mathrm e}^{\frac {-\sqrt {-\frac {\left (-\left (-\mathit {\_a} +x \right ) b \lambda 2 +\left (\lambda 2 y -1\right ) a \right ) \left (-\left (-\mathit {\_a} +x \right ) b \lambda 2 +\left (\lambda 2 y +1\right ) a \right )}{a^{2}}}\, a \mathit {c2} \lambda 1 -\left (\left (a y +\left (\mathit {\_a} -x \right ) b \right ) \mathit {c2} \lambda 1 \arcsin \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \lambda 2}{a}\right )+\left (\mathit {\_a} \lambda 1 \arcsin \left (\mathit {\_a} \lambda 1 \right )+\sqrt {-\mathit {\_a}^{2} \lambda 1^{2}+1}\right ) b \mathit {c1} \right ) \lambda 2}{a b \lambda 1 \lambda 2}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {\sqrt {-\lambda 2^{2} y^{2}+1}\, a \mathit {c2} \lambda 1 +\left (\sqrt {-\lambda 1^{2} x^{2}+1}\, b \mathit {c1} +\left (a \mathit {c2} y \arcsin \left (\lambda 2 y \right )+b \mathit {c1} x \arcsin \left (\lambda 1 x \right )\right ) \lambda 1 \right ) \lambda 2}{a b \lambda 1 \lambda 2}}\]
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Added April 13, 2019.
Problem Chapter 5.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arcsin ^m(\mu x) w_y = c \arcsin ^k(\nu x) w + p \arcsin ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcSin[mu*x]^m*D[w[x, y], y] == c*ArcSin[nu*x]^k*w[x,y]+p*ArcSin[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {i c \sin ^{-1}(\nu x)^k \left (\sin ^{-1}(\nu x)^2\right )^{-k} \left (\left (-i \sin ^{-1}(\nu x)\right )^k \operatorname {Gamma}\left (k+1,i \sin ^{-1}(\nu x)\right )-\left (i \sin ^{-1}(\nu x)\right )^k \operatorname {Gamma}\left (k+1,-i \sin ^{-1}(\nu x)\right )\right )}{2 a \nu }\right ) \left (\int _1^x\frac {\exp \left (\frac {i c \sin ^{-1}(\nu K[2])^k \left (\sin ^{-1}(\nu K[2])^2\right )^{-k} \left (\left (i \sin ^{-1}(\nu K[2])\right )^k \operatorname {Gamma}\left (k+1,-i \sin ^{-1}(\nu K[2])\right )-\left (-i \sin ^{-1}(\nu K[2])\right )^k \operatorname {Gamma}\left (k+1,i \sin ^{-1}(\nu K[2])\right )\right )}{2 a \nu }\right ) p \sin ^{-1}\left (\beta \left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^n}{a}dK[2]+c_1\left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arcsin(mu*x)^m*diff(w(x,y),y) = c*arcsin(nu*x)^k*w(x,y)+p*arcsin(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (-\arcsin \left (\frac {\left (-\left (\arcsin \left (\mathit {\_f} \mu \right )^{m}-\frac {\LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_f} \mu \right )\right )}{\sqrt {\arcsin \left (\mathit {\_f} \mu \right )}}\right ) \sqrt {-\mathit {\_f}^{2} \mu ^{2}+1}\, b 2^{m} 2^{-m} \arcsin \left (\mathit {\_f} \mu \right )+\left (-\mathit {\_f} b m 2^{m} 2^{-m} \LommelS 1\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mathit {\_f} \mu \right )\right ) \sqrt {\arcsin \left (\mathit {\_f} \mu \right )}-\frac {\mathit {\_f} b 2^{m} 2^{-m} \LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_f} \mu \right )\right )}{\sqrt {\arcsin \left (\mathit {\_f} \mu \right )}}-\left (m +1\right ) a y +\left (m +1\right ) a \left (\int \frac {b \arcsin \left (\mu x \right )^{m}}{a}d x \right )\right ) \mu \right ) \left (\mathit {\_f} \mu +1\right ) \left (\mathit {\_f} \mu -1\right ) \beta }{\left (m +1\right ) \left (\mathit {\_f}^{2} \mu ^{2}-1\right ) a \mu }\right )\right )^{n} {\mathrm e}^{\frac {\left (-\left (k \LommelS 1\left (k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mathit {\_f} \nu \right )\right ) \sqrt {\arcsin \left (\mathit {\_f} \nu \right )}+\frac {\LommelS 1\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_f} \nu \right )\right )}{\sqrt {\arcsin \left (\mathit {\_f} \nu \right )}}\right ) \mathit {\_f} \nu +\left (-\arcsin \left (\mathit {\_f} \nu \right )^{k}+\frac {\LommelS 1\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_f} \nu \right )\right )}{\sqrt {\arcsin \left (\mathit {\_f} \nu \right )}}\right ) \sqrt {-\mathit {\_f}^{2} \nu ^{2}+1}\, \arcsin \left (\mathit {\_f} \nu \right )\right ) \left (\mathit {\_f} \nu -1\right ) \left (\mathit {\_f} \nu +1\right ) c 2^{k} 2^{-k}}{\left (k +1\right ) \left (\mathit {\_f}^{2} \nu ^{2}-1\right ) a \nu }}}{a}d\mathit {\_f} +\mathit {\_F1} \left (\frac {-\left (-\LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu x \right )\right ) \arcsin \left (\mu x \right )+\arcsin \left (\mu x \right )^{m +\frac {3}{2}}\right ) \sqrt {-\mu ^{2} x^{2}+1}\, b +\left (-b m x \LommelS 1\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mu x \right )\right ) \arcsin \left (\mu x \right )-b x \LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu x \right )\right )+\left (m +1\right ) a y \sqrt {\arcsin \left (\mu x \right )}\right ) \mu }{\left (m +1\right ) a \mu \sqrt {\arcsin \left (\mu x \right )}}\right )\right ) {\mathrm e}^{\int \frac {c \arcsin \left (\nu x \right )^{k}}{a}d x}\]
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Added April 13, 2019.
Problem Chapter 5.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arcsin ^m(\mu x) w_y = c \arcsin ^k(\nu y) w + p \arcsin ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcSin[mu*x]^m*D[w[x, y], y] == c*ArcSin[nu*y]^k*w[x,y]+p*ArcSin[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin ^{-1}\left (\nu \left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \sin ^{-1}\left (\nu \left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) p \sin ^{-1}(\beta K[3])^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arcsin(mu*x)^m*diff(w(x,y),y) = c*arcsin(nu*y)^k*w(x,y)+p*arcsin(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \arcsin \left (\mathit {\_f} \beta \right )^{n} {\mathrm e}^{-\frac {c \left (\int \left (-\arcsin \left (\frac {\left (-\left (\arcsin \left (\mathit {\_f} \mu \right )^{m}-\frac {\LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_f} \mu \right )\right )}{\sqrt {\arcsin \left (\mathit {\_f} \mu \right )}}\right ) \sqrt {-\mathit {\_f}^{2} \mu ^{2}+1}\, b 2^{m} 2^{-m} \arcsin \left (\mathit {\_f} \mu \right )+\left (-\mathit {\_f} b m 2^{m} 2^{-m} \LommelS 1\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mathit {\_f} \mu \right )\right ) \sqrt {\arcsin \left (\mathit {\_f} \mu \right )}-\frac {\mathit {\_f} b 2^{m} 2^{-m} \LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_f} \mu \right )\right )}{\sqrt {\arcsin \left (\mathit {\_f} \mu \right )}}-\left (m +1\right ) a y +\left (m +1\right ) a \left (\int \frac {b \arcsin \left (\mu x \right )^{m}}{a}d x \right )\right ) \mu \right ) \left (\mathit {\_f} \mu +1\right ) \left (\mathit {\_f} \mu -1\right ) \nu }{\left (m +1\right ) \left (\mathit {\_f}^{2} \mu ^{2}-1\right ) a \mu }\right )\right )^{k}d \mathit {\_f} \right )}{a}}}{a}d\mathit {\_f} +\mathit {\_F1} \left (\frac {-\left (-\LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu x \right )\right ) \arcsin \left (\mu x \right )+\arcsin \left (\mu x \right )^{m +\frac {3}{2}}\right ) \sqrt {-\mu ^{2} x^{2}+1}\, b +\left (-b m x \LommelS 1\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mu x \right )\right ) \arcsin \left (\mu x \right )-b x \LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu x \right )\right )+\left (m +1\right ) a y \sqrt {\arcsin \left (\mu x \right )}\right ) \mu }{\left (m +1\right ) a \mu \sqrt {\arcsin \left (\mu x \right )}}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (-\arcsin \left (\frac {\left (-\left (\arcsin \left (\mathit {\_b} \mu \right )^{m}-\frac {\LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_b} \mu \right )\right )}{\sqrt {\arcsin \left (\mathit {\_b} \mu \right )}}\right ) \sqrt {-\mathit {\_b}^{2} \mu ^{2}+1}\, b 2^{m} 2^{-m} \arcsin \left (\mathit {\_b} \mu \right )+\left (-\mathit {\_b} b m 2^{m} 2^{-m} \LommelS 1\left (m +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mathit {\_b} \mu \right )\right ) \sqrt {\arcsin \left (\mathit {\_b} \mu \right )}-\frac {\mathit {\_b} b 2^{m} 2^{-m} \LommelS 1\left (m +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_b} \mu \right )\right )}{\sqrt {\arcsin \left (\mathit {\_b} \mu \right )}}-\left (m +1\right ) a y +\left (m +1\right ) a \left (\int \frac {b \arcsin \left (\mu x \right )^{m}}{a}d x \right )\right ) \mu \right ) \left (\mathit {\_b} \mu +1\right ) \left (\mathit {\_b} \mu -1\right ) \nu }{\left (m +1\right ) \left (\mathit {\_b}^{2} \mu ^{2}-1\right ) a \mu }\right )\right )^{k}}{a}d\mathit {\_b}}\]
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