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]];
 

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

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( k+1 \right ) \lambda } \left ( a{2}^{k}{2}^{-k} \left ( \arcsin \left ( \lambda \,x \right ) \LommelS 1 \left ( k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,x \right ) \right ) - \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k} \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{{\frac {3}{2}}} \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}- \left ( ax{2}^{k}{2}^{-k}\LommelS 1 \left ( k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,x \right ) \right ) +a\arcsin \left ( \lambda \,x \right ) \LommelS 1 \left ( k+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \lambda \,x \right ) \right ) kx{2}^{k}{2}^{-k}- \left ( -2\,a \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k}x{2}^{k}{2}^{-1-k}+a \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k}x{2}^{k}{2}^{-k}- \left ( k+1 \right ) \left ( bx-y \right ) \right ) \sqrt {\arcsin \left ( \lambda \,x \right ) } \right ) \lambda \right ) {\frac {1}{\sqrt {\arcsin \left ( \lambda \,x \right ) }}}} \right ) \]

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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]];
 

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

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \arcsin \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]

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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]];
 

Failed

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {ax+by}{b}}}\! \left ( k \left ( \arcsin \left ( {\it \_a}\,b+c \right ) \right ) ^{n}b+a \right ) ^{-1}{d{\it \_a}}b+x \right ) \]

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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 ) ={\it \_F1} \left ( -{\frac { \left ( \lambda \,x+1 \right ) \left ( \lambda \,x-1 \right ) }{ \left ( k+1 \right ) \lambda \, \left ( {\lambda }^{2}{x}^{2}-1 \right ) a\mu \, \left ( -1+n \right ) } \left ( a{2}^{-k}\arcsin \left ( \lambda \,x \right ) \mu \,{2}^{k} \left ( -1+n \right ) \left ( -{\LommelS 1 \left ( k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,x \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \lambda \,x \right ) }}}}+ \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k} \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}- \left ( {{2}^{-n}{2}^{n} \left ( k+1 \right ) \left ( \arcsin \left ( \mu \,y \right ) \LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,y \right ) \right ) - \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{-n} \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{{\frac {3}{2}}} \right ) \sqrt {-{\mu }^{2}{y}^{2}+1}{\frac {1}{\sqrt {\arcsin \left ( \mu \,y \right ) }}}}+ \left ( -{{2}^{-n}y{2}^{n} \left ( k+1 \right ) \LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,y \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,y \right ) }}}}+{2}^{-n}\sqrt {\arcsin \left ( \mu \,y \right ) }ny{2}^{n} \left ( k+1 \right ) \LommelS 1 \left ( -n+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \mu \,y \right ) \right ) -{a{2}^{-k}x{2}^{k} \left ( -1+n \right ) \LommelS 1 \left ( k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,x \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \lambda \,x \right ) }}}}-a{2}^{-k}k\sqrt {\arcsin \left ( \lambda \,x \right ) }x{2}^{k} \left ( -1+n \right ) \LommelS 1 \left ( k+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \lambda \,x \right ) \right ) -2\,{2}^{-n} \left ( {2}^{-1+n}-1/2\,{2}^{n} \right ) y \left ( k+1 \right ) \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{-n}+a \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k}x{2}^{k} \left ( -1+n \right ) \left ( {2}^{-k}-2\,{2}^{-1-k} \right ) \right ) \mu \right ) \lambda \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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a+y} \left ( \left ( -a-y \right ) \int \!{{\rm e}^{-2\,{\frac { \left ( x-1 \right ) \left ( x+1 \right ) }{ \left ( n+1 \right ) {x}^{2}-1-n} \left ( 1/2\,{2}^{n}\lambda \,{2}^{-n}\arcsin \left ( x \right ) \left ( {\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\arcsin \left ( x \right ) }}}- \left ( \arcsin \left ( x \right ) \right ) ^{n} \right ) \sqrt {-{x}^{2}+1}+ \left ( -1/2\,{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) {2}^{n}\lambda \,{2}^{-n}}{\sqrt {\arcsin \left ( x \right ) }}}-1/2\,\sqrt {\arcsin \left ( x \right ) }\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) n{2}^{n}\lambda \,{2}^{-n}-{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{n}\lambda +1/2\,{2}^{n}\lambda \,{2}^{-n} \left ( \arcsin \left ( x \right ) \right ) ^{n}+a \left ( n+1 \right ) \right ) x \right ) }}}\,{\rm d}x-{{\rm e}^{-2\,{\frac { \left ( x-1 \right ) \left ( x+1 \right ) }{ \left ( n+1 \right ) {x}^{2}-1-n} \left ( 1/2\,{2}^{n}\lambda \,{2}^{-n}\arcsin \left ( x \right ) \left ( {\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\arcsin \left ( x \right ) }}}- \left ( \arcsin \left ( x \right ) \right ) ^{n} \right ) \sqrt {-{x}^{2}+1}+ \left ( -1/2\,{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) {2}^{n}\lambda \,{2}^{-n}}{\sqrt {\arcsin \left ( x \right ) }}}-1/2\,\sqrt {\arcsin \left ( x \right ) }\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) n{2}^{n}\lambda \,{2}^{-n}-{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{n}\lambda +1/2\,{2}^{n}\lambda \,{2}^{-n} \left ( \arcsin \left ( x \right ) \right ) ^{n}+a \left ( n+1 \right ) \right ) x \right ) }}} \right ) } \right ) \]

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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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\exp \left (-\int _1^x-\lambda \text {Arcsin}(K[5])^n K[5]dK[5]\right )}{x^2 y+x}-\int _1^x\frac {\exp \left (-\int _1^{K[6]}-\lambda \text {Arcsin}(K[5])^n K[5]dK[5]\right )}{K[6]^2}dK[6]\right )\right \}\right \}\]

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]

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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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{{x}^{k+1}y-1} \left ( -{{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}{x}^{k+1}\lambda \,x-2\,k-2}{x}}\,{\rm d}x}}{x}^{k+1}+\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arcsin \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x \left ( {x}^{k+1}y-1 \right ) \left ( k+1 \right ) \right ) } \right ) \]

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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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b+y} \left ( -\lambda \, \left ( b+y \right ) \int \! \left ( \arcsin \left ( x \right ) \right ) ^{n}{{\rm e}^{{\frac { \left ( x-1 \right ) \left ( x+1 \right ) }{ \left ( n+1 \right ) {x}^{2}-n-1} \left ( -2\,b{2}^{n}\lambda \,{2}^{-n}\arcsin \left ( x \right ) \left ( -{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\arcsin \left ( x \right ) }}}+ \left ( \arcsin \left ( x \right ) \right ) ^{n} \right ) \sqrt {-{x}^{2}+1}+ \left ( -2\,{\frac {\lambda \,b{2}^{n}{2}^{-n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\arcsin \left ( x \right ) }}}-2\,\sqrt {\arcsin \left ( x \right ) }bn{2}^{n}\lambda \,{2}^{-n}\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) -4\,{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n}b{2}^{n}\lambda +2\, \left ( \arcsin \left ( x \right ) \right ) ^{n}b{2}^{n}\lambda \,{2}^{-n}+a \left ( n+1 \right ) \right ) x \right ) }}}\,{\rm d}x-{{\rm e}^{{\frac { \left ( x-1 \right ) \left ( x+1 \right ) }{ \left ( n+1 \right ) {x}^{2}-n-1} \left ( -2\,b{2}^{n}\lambda \,{2}^{-n}\arcsin \left ( x \right ) \left ( -{\frac {\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\arcsin \left ( x \right ) }}}+ \left ( \arcsin \left ( x \right ) \right ) ^{n} \right ) \sqrt {-{x}^{2}+1}+ \left ( -2\,{\frac {\lambda \,b{2}^{n}{2}^{-n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\arcsin \left ( x \right ) }}}-2\,\sqrt {\arcsin \left ( x \right ) }bn{2}^{n}\lambda \,{2}^{-n}\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) -4\,{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n}b{2}^{n}\lambda +2\, \left ( \arcsin \left ( x \right ) \right ) ^{n}b{2}^{n}\lambda \,{2}^{-n}+a \left ( n+1 \right ) \right ) x \right ) }}} \right ) } \right ) \]

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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