6.2.20 7.1

6.2.20.1 [703] problem number 1

problem number 703

Added January 20, 2019.

Problem 2.7.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \arcsin ^k(\lambda x)+b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*ArcSin[lambda*x]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {i a \arcsin (\lambda x)^k \left (\arcsin (\lambda x)^2\right )^{-k} \left ((i \arcsin (\lambda x))^k \Gamma (k+1,-i \arcsin (\lambda x))-(-i \arcsin (\lambda x))^k \Gamma (k+1,i \arcsin (\lambda x))\right )}{2 \lambda }-b x+y\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ (a*arcsin(lambda*x)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {a \left (-\arcsin \left (\lambda x \right ) \operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )+\arcsin \left (\lambda x \right )^{k +\frac {3}{2}}\right ) \sqrt {-\lambda ^{2} x^{2}+1}+\left (a x \operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )+a \operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) k x \arcsin \left (\lambda x \right )+\sqrt {\arcsin \left (\lambda x \right )}\, \left (k +1\right ) \left (b x -y \right )\right ) \lambda }{\sqrt {\arcsin \left (\lambda x \right )}\, \left (k +1\right ) \lambda }\right )\]

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6.2.20.2 [704] problem number 2

problem number 704

Added January 20, 2019.

Problem 2.7.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \arcsin ^k(\lambda y)+b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*ArcSin[lambda*y]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\frac {1}{a \arcsin (\lambda K[1])^k+b}dK[1]-x\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ (a*arcsin(lambda*y)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\int \frac {1}{a \arcsin \left (\lambda y \right )^{k}+b}d y +x \right )\]

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6.2.20.3 [705] problem number 3

problem number 705

Added January 20, 2019.

Problem 2.7.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + k \arcsin ^n(a x + b y+c) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + k*Arcsin[a*x + b*y + c]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {b c_1}{b k \text {Arcsin}(c+a x+b K[6172])^n+a}-\int _1^x\frac {a b^2 k n \text {Arcsin}(c+a K[1]+b K[6172])^{n-1} c_1 \text {Arcsin}'(c+a K[1]+b K[6172])}{\left (b k \text {Arcsin}(c+a K[1]+b K[6172])^n+a\right )^2}dK[1]\right )dK[6172]+\int _1^x\frac {b k \text {Arcsin}(c+b y+a K[1])^n c_1}{b k \text {Arcsin}(c+b y+a K[1])^n+a}dK[1]+c_2\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ k*arcsin(a*x + b*y+c)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\int _{}^{\frac {a x +b y}{b}}\frac {1}{k \arcsin \left (\textit {\_a} b +c \right )^{n} b +a}d \textit {\_a} b +x \right )\]

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6.2.20.4 [706] problem number 4

problem number 706

Added January 20, 2019.

Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \arcsin ^k(\lambda x) \arcsin ^n(\mu y) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Arcsin[lambda*x]^k*Arcsin[mu*y]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\text {Arcsin}(\mu K[1])^{-n}dK[1]-\int _1^xa \text {Arcsin}(\lambda K[2])^kdK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+a*arcsin(lambda*x)^k*arcsin(mu*y)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left (\sqrt {\arcsin \left (\lambda x \right )}\, \left (k +1\right ) \left (-\operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu y \right )\right ) \arcsin \left (\mu y \right )+\arcsin \left (\mu y \right )^{-n +\frac {3}{2}}\right ) \sqrt {-\mu ^{2} y^{2}+1}+\mu \left (\sqrt {\arcsin \left (\lambda x \right )}\, y \left (k +1\right ) \operatorname {LommelS1}\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mu y \right )\right )-\sqrt {\arcsin \left (\lambda x \right )}\, n y \arcsin \left (\mu y \right ) \left (k +1\right ) \operatorname {LommelS1}\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mu y \right )\right )+a \sqrt {\arcsin \left (\mu y \right )}\, x \left (n -1\right ) \left (k \operatorname {LommelS1}\left (k +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) \arcsin \left (\lambda x \right )+\operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )\right )\right )\right ) \lambda \sqrt {-\lambda ^{2} x^{2}+1}-a \sqrt {\arcsin \left (\mu y \right )}\, \mu \left (\lambda x -1\right ) \left (\lambda x +1\right ) \left (n -1\right ) \left (-\arcsin \left (\lambda x \right ) \operatorname {LommelS1}\left (k +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )+\arcsin \left (\lambda x \right )^{k +\frac {3}{2}}\right )}{\sqrt {\arcsin \left (\lambda x \right )}\, \sqrt {-\lambda ^{2} x^{2}+1}\, \sqrt {\arcsin \left (\mu y \right )}\, \mu \left (n -1\right ) a \lambda \left (k +1\right )}\right )\]

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6.2.20.5 [707] problem number 5

problem number 707

Added January 20, 2019.

Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+ \lambda (\arcsin x)^n y -a^2 + a \lambda ( \arcsin x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*Arcsin[x]^n*y - a^2 + a*lambda*Arcsin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(y^2+ lambda*arcsin(x)^n*y -a^2 + a *lambda*arcsin(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-a -y \right ) \int {\mathrm e}^{-\frac {2 \left (\frac {\lambda \left (\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right ) \arcsin \left (x \right )-\arcsin \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}}{2}+x \left (-\frac {\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right ) \lambda }{2}-\frac {\arcsin \left (x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right ) \lambda n}{2}+a \sqrt {\arcsin \left (x \right )}\, \left (1+n \right )\right )\right )}{\sqrt {\arcsin \left (x \right )}\, \left (1+n \right )}}d x -{\mathrm e}^{-\frac {2 \left (\frac {\lambda \left (\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right ) \arcsin \left (x \right )-\arcsin \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}}{2}+x \left (-\frac {\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right ) \lambda }{2}-\frac {\arcsin \left (x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right ) \lambda n}{2}+a \sqrt {\arcsin \left (x \right )}\, \left (1+n \right )\right )\right )}{\sqrt {\arcsin \left (x \right )}\, \left (1+n \right )}}}{a +y}\right )\]

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6.2.20.6 [708] problem number 6

problem number 708

Added January 20, 2019.

Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+ \lambda x (\arcsin x)^n y + \lambda ( \arcsin y)^n \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*x*Arcsin[x]^n*y + lambda*Arcsin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\exp \left (-\int _1^x-\lambda \text {Arcsin}(K[1])^n K[1]dK[1]\right )}{x^2 y+x}-\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\lambda \text {Arcsin}(K[1])^n K[1]dK[1]\right )}{K[2]^2}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( y^2+ lambda*x*arcsin(x)^n*y + lambda*arcsin(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\int \frac {{\mathrm e}^{\frac {4 \lambda \left (\left (\left (\frac {n \sin \left (2 \arcsin \left (x \right )\right )}{4}+\arcsin \left (x \right ) \left (n \,x^{2}+2 x^{2}-1\right )\right ) 2^{\frac {1}{2}-n}-\frac {2^{\frac {3}{2}-n} \left (n \arcsin \left (x \right )-\sin \left (2 \arcsin \left (x \right )\right )\right )}{4}\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 2 \arcsin \left (x \right )\right )-\frac {\sin \left (2 \arcsin \left (x \right )\right ) \left (2^{\frac {3}{2}-n} \arcsin \left (x \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 2 \arcsin \left (x \right )\right )-4 \arcsin \left (x \right )^{n +\frac {3}{2}}\right )}{4}\right )}{\sqrt {\arcsin \left (x \right )}\, \left (8 n +16\right )}}}{x^{2}}d x \left (y \,x^{2}+x \right )+{\mathrm e}^{\frac {4 \lambda \left (\left (\left (\frac {n \sin \left (2 \arcsin \left (x \right )\right )}{4}+\arcsin \left (x \right ) \left (n \,x^{2}+2 x^{2}-1\right )\right ) 2^{\frac {1}{2}-n}-\frac {2^{\frac {3}{2}-n} \left (n \arcsin \left (x \right )-\sin \left (2 \arcsin \left (x \right )\right )\right )}{4}\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 2 \arcsin \left (x \right )\right )-\frac {\sin \left (2 \arcsin \left (x \right )\right ) \left (2^{\frac {3}{2}-n} \arcsin \left (x \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 2 \arcsin \left (x \right )\right )-4 \arcsin \left (x \right )^{n +\frac {3}{2}}\right )}{4}\right )}{\sqrt {\arcsin \left (x \right )}\, \left (8 n +16\right )}}}{x \left (x y +1\right )}\right )\]

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6.2.20.7 [709] problem number 7

problem number 709

Added January 20, 2019.

Problem 2.7.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x - \left ( (k+1) x^k y^2 - \lambda (\arcsin x)^n (x^{k+1} y-1) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*Arcsin[x]^n*(x^(k + 1)*y - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)-(  (k+1)*x^k*y^2 - lambda*arcsin(x)^n*(x^(k+1)*y-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {x^{k +1} {\mathrm e}^{\int \frac {\arcsin \left (x \right )^{n} x^{k +1} \lambda x -2 k -2}{x}d x}-\int x^{-k -2} {\mathrm e}^{\lambda \int x^{k +1} \arcsin \left (x \right )^{n}d x}d x \left (x^{k +1} y -1\right ) \left (k +1\right )}{x^{k +1} y -1}\right )\]

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6.2.20.8 [710] problem number 8

problem number 710

Added January 20, 2019.

Problem 2.7.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arcsin x)^n y^2 + a y+ a b -b^2 \lambda (\arcsin x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Arcsin[x]^n*y^2 + a*y + a*b - b^2*lambda*Arcsin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arcsin(x)^n*y^2 + a*y+ a*b -b^2 * lambda*arcsin(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\lambda \left (b +y \right ) \int \arcsin \left (x \right )^{n} {\mathrm e}^{\frac {-2 b \lambda \left (-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right ) \arcsin \left (x \right )+\arcsin \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}+\left (-2 b \lambda \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )-2 \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right ) b n \lambda \arcsin \left (x \right )+a \sqrt {\arcsin \left (x \right )}\, \left (1+n \right )\right ) x}{\sqrt {\arcsin \left (x \right )}\, \left (1+n \right )}}d x -{\mathrm e}^{\frac {-2 b \lambda \left (-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right ) \arcsin \left (x \right )+\arcsin \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}+\left (-2 b \lambda \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )-2 \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right ) b n \lambda \arcsin \left (x \right )+a \sqrt {\arcsin \left (x \right )}\, \left (1+n \right )\right ) x}{\sqrt {\arcsin \left (x \right )}\, \left (1+n \right )}}}{b +y}\right )\]

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6.2.20.9 [711] problem number 9

problem number 711

Added January 29, 2019.

Problem 2.7.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arcsin x)^n y^2 - b \lambda x^m (\arcsin x)^n y+ b m x^{m-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Arcsin[x]^n*y^2 - b*lambda*x^m*ArcSin[x]^n*y + b*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arcsin(x)^n*y^2 - b*lambda*x^m*arcsin(x)^n*y+b*m*x^(m-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.20.10 [712] problem number 10

problem number 712

Added January 29, 2019.

Problem 2.7.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arcsin x)^n y^2 + b m x^{m-1} - \lambda b^2 x^{2 m} (\arcsin x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcSin[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcSin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arcsin(x)^n*y^2 + b*m*x^(m-1) - lambda*b^2*x^(2*m)*arcsin(x)^n  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.20.11 [713] problem number 11

problem number 713

Added January 29, 2019.

Problem 2.7.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arcsin x)^n (y - a x^m -b)^2 + a m x^{m-1} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcSin[x]^n*(y - a*x^m - b)^2 + a*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {1}{a x^m+b-y}-\frac {1}{2} i \lambda (i \arcsin (x))^n \arcsin (x)^n \left (\arcsin (x)^2\right )^{-n} \Gamma (n+1,-i \arcsin (x))+\frac {1}{2} i \lambda (-i \arcsin (x))^n \arcsin (x)^n \left (\arcsin (x)^2\right )^{-n} \Gamma (n+1,i \arcsin (x))\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arcsin(x)^n*(y - a*x^m -b)^2 + a*m*x^(m-1)  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\lambda \left (-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right ) \arcsin \left (x \right )+\arcsin \left (x \right )^{n +\frac {3}{2}}\right ) \left (a \,x^{m}+b -y \right ) \sqrt {-x^{2}+1}+\left (a \,x^{1+m}+x \left (b -y \right )\right ) \lambda \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )+n \left (a \,x^{1+m}+x \left (b -y \right )\right ) \arcsin \left (x \right ) \lambda \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right )-\sqrt {\arcsin \left (x \right )}\, \left (n +1\right )}{\sqrt {\arcsin \left (x \right )}\, \left (a \,x^{m}+b -y \right ) \left (n +1\right )}\right )\]

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6.2.20.12 [714] problem number 12

problem number 714

Added January 29, 2019.

Problem 2.7.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( \lambda (\arcsin x)^n y^2 +k y+ \lambda b^2 x^{2 k} (\arcsin x)^n \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (lambda*ArcSin[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcSin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x\lambda \arcsin (K[1])^n K[1]^{k-1}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  x*diff(w(x,y),x)+( lambda*arcsin(x)^n*y^2 +k*y+ lambda*b^2*x^(2*k)*arcsin(x)^n  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (b \lambda \int x^{k -1} \arcsin \left (x \right )^{n}d x -\arctan \left (\frac {x^{-k} y}{b}\right )\right )\]

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