6.4.20 7.1

6.4.20.1 [1140] Problem 1

problem number 1140

Added March 9, 2019.

Problem Chapter 4.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = \left ( c \arcsin (\frac {x}{\lambda } + k \arcsin (\frac {y}{\beta } ) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (c*ArcSin[x/lambda] + k*ArcSin[y/beta])*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {\frac {k \left (\sqrt {a^2 \left (\beta ^2-y^2\right )} \arctan \left (\frac {a y}{\sqrt {a^2 \left (\beta ^2-y^2\right )}}\right ) (a y-b x)+a^2 \left (\beta ^2-y^2\right )\right )}{b \beta \sqrt {1-\frac {y^2}{\beta ^2}}}+a k x \arcsin \left (\frac {y}{\beta }\right )+a c x \arcsin \left (\frac {x}{\lambda }\right )+a c \lambda \sqrt {1-\frac {x^2}{\lambda ^2}}}{a^2}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*arcsin(x/lambda)+k*arcsin(y/beta))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -b x}{a}\right ) {\mathrm e}^{\frac {y \arcsin \left (\frac {y}{\beta }\right ) k a +\sqrt {\frac {\beta ^{2}-y^{2}}{\beta ^{2}}}\, k \beta a +\sqrt {\frac {\lambda ^{2}-x^{2}}{\lambda ^{2}}}\, b c \lambda +\arcsin \left (\frac {x}{\lambda }\right ) b c x}{a b}}\]

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6.4.20.2 [1141] Problem 2

problem number 1141

Added March 9, 2019.

Problem Chapter 4.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c \arcsin (\lambda x+\beta y) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcSin[lambda*x + beta*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c \left ((\beta y+\lambda x) \arcsin (\beta y+\lambda x)+\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}\right )}{a \lambda +b \beta }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*arcsin(lambda*x+beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -x b}{a}\right ) {\mathrm e}^{\frac {\left (\arcsin \left (\beta y +\lambda x \right ) \beta y +\arcsin \left (\beta y +\lambda x \right ) \lambda x +\sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -x^{2} \lambda ^{2}+1}\right ) c}{a \lambda +b \beta }}\]

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6.4.20.3 [1142] Problem 3

problem number 1142

Added March 9, 2019.

Problem Chapter 4.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = a x \arcsin (\lambda x+\beta y) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == a*x*ArcSin[lambda*x + beta*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {a \left (\arcsin (\beta y+\lambda x) \left (a \left (-2 \beta ^2 y^2+2 \lambda ^2 x^2-1\right )+4 b \beta x (\beta y+\lambda x)\right )+\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1} (-3 a \beta y+a \lambda x+4 b \beta x)\right )}{4 (a \lambda +b \beta )^2}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arcsin(lambda*x+beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -x b}{a}\right ) {\mathrm e}^{\frac {a \left (\left (\frac {\left (-3 \beta y +\lambda x \right ) a}{2}+2 b \beta x \right ) \sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -x^{2} \lambda ^{2}+1}+\left (\left (x^{2} \lambda ^{2}-\beta ^{2} y^{2}-\frac {1}{2}\right ) a +2 b \beta x \left (\beta y +\lambda x \right )\right ) \arcsin \left (\beta y +\lambda x \right )\right )}{2 \left (a \lambda +b \beta \right )^{2}}}\]

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6.4.20.4 [1143] Problem 4

problem number 1143

Added March 9, 2019.

Problem Chapter 4.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \arcsin ^n(\lambda x)w_y = \left ( c \arcsin ^m(\mu x) + s \arcsin ^k(\beta y) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*ArcSin[lambda*x]^n*D[w[x, y], y] == (c*ArcSin[mu*x]^m + s*ArcSin[beta*y]^k)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac {b \arcsin (\lambda K[1])^n}{a}dK[1]\right ) \exp \left (\int _1^x\frac {s \arcsin \left (\beta \left (y-\int _1^x\frac {b \arcsin (\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac {b \arcsin (\lambda K[1])^n}{a}dK[1]\right )\right ){}^k+c \arcsin (\mu K[2])^m}{a}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*arcsin(lambda*x)^n*diff(w(x,y),y) =(c*arcsin(mu*x)^m+s*arcsin(beta*y)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-b \left (-\arcsin \left (\lambda x \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )+\arcsin \left (\lambda x \right )^{n +\frac {3}{2}}\right ) \sqrt {-\lambda ^{2} x^{2}+1}+\left (-b x \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )-\operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) b n x \arcsin \left (\lambda x \right )+a \sqrt {\arcsin \left (\lambda x \right )}\, y \left (n +1\right )\right ) \lambda }{\sqrt {\arcsin \left (\lambda x \right )}\, \left (n +1\right ) \lambda a}\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (c \arcsin \left (\mu \textit {\_b} \right )^{m}+s {\arcsin \left (\frac {\beta \left (\lambda \left (\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda \textit {\_b} \right )\right ) \textit {\_b} b +\arcsin \left (\lambda \textit {\_b} \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda \textit {\_b} \right )\right ) b n \textit {\_b} +\sqrt {\arcsin \left (\lambda \textit {\_b} \right )}\, \left (n +1\right ) \left (a y -b \int \arcsin \left (\lambda x \right )^{n}d x \right )\right ) \sqrt {-\textit {\_b}^{2} \lambda ^{2}+1}+b \left (\lambda \textit {\_b} -1\right ) \left (\lambda \textit {\_b} +1\right ) \left (\arcsin \left (\lambda \textit {\_b} \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda \textit {\_b} \right )\right )-\arcsin \left (\lambda \textit {\_b} \right )^{n +\frac {3}{2}}\right )\right )}{\sqrt {-\textit {\_b}^{2} \lambda ^{2}+1}\, \sqrt {\arcsin \left (\lambda \textit {\_b} \right )}\, \left (n +1\right ) a \lambda }\right )}^{k}\right )d \textit {\_b}}{a}}\]

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6.4.20.5 [1144] Problem 5

problem number 1144

Added March 9, 2019.

Problem Chapter 4.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \arcsin ^n(\lambda y)w_y = \left ( c \arcsin ^m(\mu x) + s \arcsin ^k(\beta y) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*ArcSin[lambda*y]^n*D[w[x, y], y] == (c*ArcSin[mu*x]^m + s*ArcSin[beta*y]^k)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\arcsin (\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right ) \exp \left (\int _1^y\frac {\arcsin (\lambda K[2])^{-n} \left (s \arcsin (\beta K[2])^k+c \arcsin \left (\frac {\mu \left (b x-a \int _1^y\arcsin (\lambda K[1])^{-n}dK[1]+a \int _1^{K[2]}\arcsin (\lambda K[1])^{-n}dK[1]\right )}{b}\right ){}^m\right )}{b}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*arcsin(lambda*y)^n*diff(w(x,y),y) =(c*arcsin(mu*x)^m+s*arcsin(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 ) = f_{1} \left (\frac {a \left (-\operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )+\arcsin \left (\lambda y \right )^{-n +\frac {3}{2}}\right ) \sqrt {-\lambda ^{2} y^{2}+1}-\left (-\operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right ) a y +\operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right ) n a y -\sqrt {\arcsin \left (\lambda y \right )}\, b x \left (n -1\right )\right ) \lambda }{\sqrt {\arcsin \left (\lambda y \right )}\, \left (n -1\right ) \lambda b}\right ) {\mathrm e}^{\frac {\int _{}^{y}\left (c {\left (-\arcsin \left (\frac {\mu \left (\left (a \arcsin \left (\lambda \textit {\_b} \right )^{n} \operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda \textit {\_b} \right )\right ) \textit {\_b} -a \arcsin \left (\lambda \textit {\_b} \right )^{1+n} \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda \textit {\_b} \right )\right ) n \textit {\_b} +\arcsin \left (\lambda \textit {\_b} \right )^{n +\frac {1}{2}} \left (n -1\right ) \left (a \int \arcsin \left (\lambda y \right )^{-n}d y -b x \right )\right ) \lambda \sqrt {-\textit {\_b}^{2} \lambda ^{2}+1}-a \left (\lambda \textit {\_b} -1\right ) \left (\lambda \textit {\_b} +1\right ) \left (\arcsin \left (\lambda \textit {\_b} \right )^{{3}/{2}}-\arcsin \left (\lambda \textit {\_b} \right )^{1+n} \operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda \textit {\_b} \right )\right )\right )\right ) \arcsin \left (\lambda \textit {\_b} \right )^{-n -\frac {1}{2}}}{\sqrt {-\textit {\_b}^{2} \lambda ^{2}+1}\, b \lambda \left (n -1\right )}\right )\right )}^{m}+s \arcsin \left (\beta \textit {\_b} \right )^{k}\right ) \arcsin \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{b}}\]

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