7.2.21 7.2

7.2.21.1 [717] problem number 1

problem number 717

Added January 29, 2019.

Problem 2.7.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a \arccos ^k(\lambda x) + b \right ) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( a*arccos(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'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (-b x +\frac {\sqrt {\pi }\, \left (-\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-k} \LommelS 1 \left (k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}}{\sqrt {\pi }\, \left (k +2\right )}+\frac {\sqrt {-\lambda ^{2} x^{2}+1}\, 2^{-k} \arccos \left (\lambda x \right )^{k +1}}{\sqrt {\pi }\, \left (k +2\right )}-\frac {3 \left (\frac {2 k}{3}+\frac {4}{3}\right ) \left (\lambda x \arccos \left (\lambda x \right )-\sqrt {-\lambda ^{2} x^{2}+1}\right ) 2^{-k -1} \LommelS 1 \left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )}{\sqrt {\pi }\, \left (k +2\right ) \sqrt {\arccos \left (\lambda x \right )}}\right ) a 2^{k}}{\lambda }+y \right )\]

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7.2.21.2 [718] problem number 2

problem number 718

Added January 29, 2019.

Problem 2.7.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a \arccos ^k(\lambda y) + b \right ) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( a*arccos(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'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (x -\left (\int \frac {1}{a \arccos \left (\lambda y \right )^{k}+b}d y \right )\right )\]

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7.2.21.3 [719] problem number 3

problem number 719

Added January 29, 2019.

Problem 2.7.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + k \arccos ^n(a x+b y+c) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + k*ArcCos[a*x + b*y + c]^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)+k*arccos(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'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (-b \left (\int _{}^{\frac {a x +b y}{b}}\frac {1}{b k \arccos \left (b \textit {\_a} +c \right )^{n}+a}d \textit {\_a} \right )+x \right )\]

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7.2.21.4 [720] problem number 4

problem number 720

Added January 29, 2019.

Problem 2.7.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + a \arccos ^k(\lambda x) \arccos ^n(\mu y) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+a*arccos(lambda*x)^k*arccos(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'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (n -2\right ) \left (-\frac {2 \left (k +2\right ) 2^{-k -1} \LommelS 1 \left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right )}{\sqrt {\arccos \left (\lambda x \right )}}+\left (\LommelS 1 \left (k +\frac {3}{2}, \frac {3}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}-\arccos \left (\lambda x \right )^{k +1}\right ) 2^{-k}\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, a \mu 2^{k} \sqrt {\arccos \left (\mu y \right )}+2 \left (\left (n -2\right ) \left (a x 2^{k} 2^{-k -1} \LommelS 1 \left (k +\frac {1}{2}, \frac {1}{2}, \arccos \left (\lambda x \right )\right ) \sqrt {\arccos \left (\lambda x \right )}\, \sqrt {\arccos \left (\mu y \right )}-\frac {y \LommelS 1 \left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu y \right )\right ) \arccos \left (\mu y \right )}{2}\right ) \mu +\left (\frac {\LommelS 1 \left (-n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\mu y \right )\right ) \arccos \left (\mu y \right )}{2}-\frac {\arccos \left (\mu y \right )^{-n +\frac {3}{2}}}{2}+\left (\frac {n}{2}-1\right ) \LommelS 1 \left (-n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\mu y \right )\right )\right ) \sqrt {-\mu ^{2} y^{2}+1}\right ) \left (k +2\right ) \lambda }{\left (n -2\right ) \left (k +2\right ) a \lambda \mu \sqrt {\arccos \left (\mu y \right )}}\right )\]

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7.2.21.5 [721] problem number 5

problem number 721

Added January 29, 2019.

Problem 2.7.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( y^2+\lambda (\arccos x)^n y- a^2 + a \lambda ( \arccos x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*ArcCos[x]^n*y - a^2 + a*lambda*ArcCos[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*arccos(x)^n*y- a^2 + a*lambda*arccos(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 ) = \textit {\_F1} \left (\frac {\left (-a -y \right ) \left (\int \frac {\left (-\left (-\left (n +2\right ) n \LommelS 1 \left (n -\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )^{3}+\arccos \left (x \right )^{n} \arccos \left (x \right )^{\frac {7}{2}}-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )^{3}+\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )^{2}\right ) \lambda x +\left (-\left (n +2\right ) \lambda n \LommelS 1 \left (n -\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )^{2}-\lambda \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )^{2}+\left (n +2\right ) \lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\left (\lambda \arccos \left (x \right )^{n}+\left (n +2\right ) \left (a +y \right )\right ) \arccos \left (x \right )^{\frac {5}{2}}\right ) \sqrt {-x^{2}+1}\right ) {\mathrm e}^{-\frac {2 \left (\frac {\left (\arccos \left (x \right )^{n} \arccos \left (x \right )^{\frac {3}{2}}-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )\right ) \sqrt {-x^{2}+1}\, \lambda }{2}+\left (-\frac {\lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )}{2}+a \sqrt {\arccos \left (x \right )}\right ) \left (n +2\right ) x \right )}{\left (n +2\right ) \sqrt {\arccos \left (x \right )}}}}{\sqrt {-x^{2}+1}\, \left (n +2\right ) \left (a +y \right ) \arccos \left (x \right )^{\frac {5}{2}}}d x \right )-{\mathrm e}^{-\frac {2 \left (\frac {\left (\arccos \left (x \right )^{n} \arccos \left (x \right )^{\frac {3}{2}}-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )\right ) \sqrt {-x^{2}+1}\, \lambda }{2}+\left (-\frac {\lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )}{2}+a \sqrt {\arccos \left (x \right )}\right ) \left (n +2\right ) x \right )}{\left (n +2\right ) \sqrt {\arccos \left (x \right )}}}}{a +y}\right )\]

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7.2.21.6 [722] problem number 6

problem number 722

Added January 29, 2019.

Problem 2.7.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( y^2+\lambda x (\arccos x)^n y+ \lambda ( \arccos x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*x*ArcCos[x]^n*y + a*lambda*ArcCos[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*x*arccos(x)^n*y + a*lambda*arccos(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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7.2.21.7 [723] problem number 7

problem number 723

Added January 29, 2019.

Problem 2.7.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x - \left ( (k+1)x^k y^2 -\lambda (\arccos x)^n (x^{k+1} y-1) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*ArcCos[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*arccos(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'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-x^{k +1} {\mathrm e}^{\int \frac {\lambda x \,x^{k +1} \arccos \left (x \right )^{n}-2 k -2}{x}d x}+\left (y \,x^{k +1}-1\right ) \left (k +1\right ) \left (\int \frac {x^{-k} {\mathrm e}^{\lambda \left (\int x^{k +1} \arccos \left (x \right )^{n}d x \right )}}{x^{2}}d x \right )}{y \,x^{k +1}-1}\right )\]

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7.2.21.8 [724] problem number 8

problem number 724

Added January 29, 2019.

Problem 2.7.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (\arccos x)^n y^2+ a y+ a b - b^2 \lambda (\arccos x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + a*y + a*b - b^2*lambda*ArcCos[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*arccos(x)^n*y^2+ a*y+ a*b - b^2*lambda*arccos(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 ) = \textit {\_F1} \left (\frac {\left (-b -y \right ) \left (\int \frac {\left (2 \left (-\left (n +2\right ) n \LommelS 1 \left (n -\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )^{2}-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )^{2}+\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {5}{2}}\right ) b x +\left (2 \left (n +2\right ) b n \LommelS 1 \left (n -\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+2 b \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )-2 \left (n +2\right ) b \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )+\left (\left (b +y \right ) n +2 y \right ) \arccos \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}\right ) \lambda \,{\mathrm e}^{\frac {2 \left (-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}+\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )\right ) \sqrt {-x^{2}+1}\, b \lambda +\left (n +2\right ) \left (-2 b \lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+a \sqrt {\arccos \left (x \right )}\right ) x}{\left (n +2\right ) \sqrt {\arccos \left (x \right )}}}}{\sqrt {-x^{2}+1}\, \left (n +2\right ) \left (b +y \right ) \arccos \left (x \right )^{\frac {3}{2}}}d x \right )-{\mathrm e}^{\frac {2 \left (-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}+\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )\right ) \sqrt {-x^{2}+1}\, b \lambda +\left (n +2\right ) \left (-2 b \lambda \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+a \sqrt {\arccos \left (x \right )}\right ) x}{\left (n +2\right ) \sqrt {\arccos \left (x \right )}}}}{b +y}\right )\]

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7.2.21.9 [725] problem number 9

problem number 725

Added January 29, 2019.

Problem 2.7.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 - b*lambda*x^m*ArcCos[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*arccos(x)^n*y^2- b*lambda*x^m*arccos(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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7.2.21.10 [726] problem number 10

problem number 726

Added January 29, 2019.

Problem 2.7.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcCos[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*arccos(x)^n*y^2+ b*m*x^(m-1) - lambda*b^2*x^(2*m)*arccos(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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7.2.21.11 [727] problem number 11

problem number 727

Added January 29, 2019.

Problem 2.7.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[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]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(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'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (\arccos \left (x \right )^{n} \arccos \left (x \right )^{\frac {3}{2}}-\LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\left (n +2\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )\right ) \left (a \,x^{m}+b -y \right ) \sqrt {-x^{2}+1}\, \lambda -\left (n +2\right ) \left (\left (a \,x^{m}+b -y \right ) \lambda x \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )-\sqrt {\arccos \left (x \right )}\right )}{\left (n +2\right ) \left (a \,x^{m}+b -y \right ) \sqrt {\arccos \left (x \right )}}\right )\]

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7.2.21.12 [728] problem number 12

problem number 728

Added January 29, 2019.

Problem 2.7.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + \left ( \lambda (\arccos x)^n y^2+ k y + \lambda b^2 x^{2 k} (\arccos x)^n \right ) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \textit {\_F1} \left (b \lambda \left (\int x^{k -1} \arccos \left (x \right )^{n}d x \right )-\arctan \left (\frac {y \,x^{-k}}{b}\right )\right )\]

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