6.2.15 6.1

6.2.15.1 [639] problem number 1

problem number 639

Added January 14, 2019.

Problem 2.6.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\lambda x)\right )}{k \lambda +\lambda }-b x+y\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-\frac {a k \lambda x}{\moverset {{\lceil \frac {k}{2}\rceil }-1}{\munderset {j =0}{\prod }}\frac {-k +2 j}{-k +2 j +1}}-b x k \lambda +y k \lambda +\left (\moverset {{\lceil \frac {k}{2}\rceil }-1}{\munderset {i =0}{\sum }}\frac {\sin \left (\lambda x \right )^{k -2 i}}{\moverset {i}{\munderset {j =1}{\prod }}\frac {-k +2 j}{-k +2 j -1}}\right ) a \cot \left (\lambda x \right )}{k \lambda }\right )\]

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6.2.15.2 [640] problem number 2

problem number 640

Added January 14, 2019.

Problem 2.6.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

contains unresolved integral

Maple ✓

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

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

contains unresolved integral

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6.2.15.3 [641] problem number 3

problem number 641

Added January 14, 2019.

Problem 2.6.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {\cos ^2(\mu y)} \sec (\mu y) \sin ^{1-n}(\mu y) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(\mu y)\right )}{\mu -\mu n}-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]

Maple ✓

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

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

contains unresolved integral

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6.2.15.4 [642] problem number 4

problem number 642

Added January 14, 2019.

Problem 2.6.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {\lambda c_1}{a \lambda \sin ^k(x+\lambda K[6467])+1}-\int _1^x\frac {a k \lambda ^2 c_1 \cos (K[1]+\lambda K[6467]) \sin ^{k-1}(K[1]+\lambda K[6467])}{\left (a \lambda \sin ^k(K[1]+\lambda K[6467])+1\right )^2}dK[1]\right )dK[6467]+\int _1^x\frac {a \lambda c_1 \sin ^k(\lambda y+K[1])}{a \lambda \sin ^k(\lambda y+K[1])+1}dK[1]+c_2\right \}\right \}\]

Maple ✓

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

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

contains unresolved integral

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6.2.15.5 [643] problem number 5

problem number 643

Added January 14, 2019.

Problem 2.6.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {e^{\frac {2 a \sin (\lambda x)}{\lambda }} (a \cos (\lambda x)+y)}{e^{\frac {2 a \sin (\lambda x)}{\lambda }} (a \cos (\lambda x)+y) \int _1^xe^{-\frac {2 a \sin (\lambda K[1])}{\lambda }}dK[1]+1}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-2 a \cos \left (\lambda x \right )-2 y \right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\lambda \cos \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{\left (\left (2 a \cos \left (\lambda x \right )+2 y \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )+\lambda \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\cos \left (\lambda x \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \lambda }\right )\]

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6.2.15.6 [644] problem number 6

problem number 644

Added January 14, 2019.

Problem 2.6.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +\left ( y^2 + a \sin (\beta x) y + a b \sin (\beta x)-b^2 \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + a*Sin[beta*x]*y + a*b*Sin[beta*x] - b^2)*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 + a*sin(beta*x)* y + a*b*sin(beta*x)-b^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {{\mathrm e}^{\frac {-2 b x \beta -a \cos \left (\beta x \right )}{\beta }}+\int {\mathrm e}^{\frac {-2 b x \beta -a \cos \left (\beta x \right )}{\beta }}d x \left (b +y \right )}{b +y}\right )\]

contains unresolved integrals

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6.2.15.7 [645] problem number 7

problem number 645

Added January 14, 2019.

Problem 2.6.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +\left ( y^2 + a x \sin ^m(b x) y + a \sin ^m(b x)\right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\int _1^x\frac {\exp \left (\frac {a \sin ^{m+1}(b K[1]) \left (\frac {2 b \cos (b K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+3}{2},\sin ^2(b K[1])\right ) K[1]}{m+1}-2^{-m-1} \sqrt {\pi } \operatorname {Gamma}(m+1) \, _3\tilde {F}_2\left (1,\frac {m+2}{2},\frac {m+2}{2};\frac {m+3}{2},\frac {m+4}{2};\sin ^2(b K[1])\right ) \sin (b K[1])\right )}{2 b^2}\right )}{K[1]^2}dK[1]-\frac {\exp \left (\frac {a \sin ^{m+1}(b x) \left (\frac {2 b x \cos (b x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+3}{2},\sin ^2(b x)\right )}{m+1}-\sqrt {\pi } 2^{-m-1} \operatorname {Gamma}(m+1) \sin (b x) \, _3\tilde {F}_2\left (1,\frac {m+2}{2},\frac {m+2}{2};\frac {m+3}{2},\frac {m+4}{2};\sin ^2(b x)\right )\right )}{2 b^2}\right )}{x^2 y+x}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {x y \int {\mathrm e}^{\int \frac {\sin \left (b x \right )^{m} x^{2} a -2}{x}d x}d x +{\mathrm e}^{\int \frac {\sin \left (b x \right )^{m} x^{2} a -2}{x}d x} x +\int {\mathrm e}^{\int \frac {\sin \left (b x \right )^{m} x^{2} a -2}{x}d x}d x}{x y +1}\right )\]

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6.2.15.8 [646] problem number 8

problem number 646

Added January 14, 2019.

Problem 2.6.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +\left (\lambda \sin (\lambda x) y^2 + \lambda \sin ^3(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + lambda*Sin[lambda*x]^3)*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*sin(lambda*x)*y^2 + lambda*sin(lambda*x)^3)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\sqrt {\pi }\, \left (y +\cos \left (\lambda x \right )\right )}{-2 \,{\mathrm e}^{\frac {\cos \left (2 \lambda x \right )}{2}+\frac {1}{2}}+\sqrt {\pi }\, \left (y +\cos \left (\lambda x \right )\right ) \operatorname {erfi}\left (\cos \left (\lambda x \right )\right )}\right )\]

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6.2.15.9 [647] problem number 9

problem number 647

Added January 14, 2019.

Problem 2.6.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-\frac {2 \left (\left (a \cos \left (\lambda x \right )+y \left (a \sin \left (\lambda x \right )-a -\lambda \right )\right ) \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )+\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \lambda \right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right ) \sqrt {-\cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2}}\, \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2 \left (\left (a \cos \left (\lambda x \right )+y \left (a \sin \left (\lambda x \right )-a -\lambda \right )\right ) \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )+\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \lambda \right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right ) \sqrt {-\cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2}}\, \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \int _{}^{\sin \left (\lambda x \right )}\frac {{\mathrm e}^{\frac {a \textit {\_a}}{\lambda }} \left (\left (\textit {\_a} -1\right ) a -\lambda \right )}{\sqrt {\textit {\_a} +1}\, \left (\textit {\_a} -1\right )^{{3}/{2}}}d \textit {\_a} +{\mathrm e}^{\frac {a \sin \left (\lambda x \right )}{\lambda }} \lambda \cos \left (\lambda x \right ) \left (a \sin \left (\lambda x \right )-a -\lambda \right )}\right )\]

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6.2.15.10 [648] problem number 10

problem number 648

Added January 14, 2019.

Problem 2.6.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\sin (\lambda x) e^{\frac {a \sin ^2(\lambda x)}{\lambda }} (y \sin (\lambda x)+\cos (\lambda x))}{\sin (\lambda x) e^{\frac {a \sin ^2(\lambda x)}{\lambda }} (y \sin (\lambda x)+\cos (\lambda x)) \int _1^xe^{-\frac {a \sin ^2(\lambda K[1])}{\lambda }} \left (\lambda \csc ^2(\lambda K[1])+a\right )dK[1]+1}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {i \left (2 \sin \left (\lambda x \right )^{2} y +\sin \left (2 \lambda x \right )\right )}{4 \lambda \left (\left (\sin \left (\lambda x \right )^{2} y +\frac {\sin \left (2 \lambda x \right )}{2}\right ) \int {\mathrm e}^{\frac {\cos \left (2 \lambda x \right ) a}{2 \lambda }} \left (a +\lambda \csc \left (\lambda x \right )^{2}\right )d x +{\mathrm e}^{\frac {\cos \left (2 \lambda x \right ) a}{2 \lambda }}\right )}\right )\]

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6.2.15.11 [649] problem number 11

problem number 649

Added January 14, 2019.

Problem 2.6.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x -\left ( (k+1) x^k y^2 - a x^{k+1}(\sin x)^m y + a (\sin x)^m \right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {(k+1) \left (y x^{k+1}-1\right )}{(k+1) \left (y x^{k+1}-1\right ) \int _1^x\exp \left (\int _1^{K[2]}-\frac {-a \sin ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )dK[2]-x \exp \left (\int _1^x-\frac {-a \sin ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )}\right )\right \}\right \}\]

Maple ✓

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

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

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6.2.15.12 [650] problem number 12

problem number 650

Added January 14, 2019.

Problem 2.6.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Sin[lambda*x + mu]^k*(y - b*x^n - c)^2 + y - b*x^n + b*n*x^(n - 1) - c)*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)+(a*sin(lambda*x + mu)^k * (y-b*x^n -c)^2 + y - b*x^n + b*n*x^(n-1) - c)*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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6.2.15.13 [651] problem number 13

problem number 651

Added January 14, 2019.

Problem 2.6.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.15.14 [652] problem number 14

problem number 652

Added January 14, 2019.

Problem 2.6.1.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a \sin (\lambda x) + b) w_x +\left ( y^2+ c \sin (\mu x) y - k^2 + c k \sin (\mu x) \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {\int \frac {{\mathrm e}^{\frac {c \int \frac {\sin \left (\mu x \right )}{a \sin \left (\lambda x \right )+b}d x \lambda \sqrt {-a^{2}+b^{2}}-4 k \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right )}{\lambda \sqrt {-a^{2}+b^{2}}}}}{a \sin \left (\lambda x \right )+b}d x \left (k +y \right )+{\mathrm e}^{\frac {c \int \frac {\sin \left (\mu x \right )}{a \sin \left (\lambda x \right )+b}d x \lambda \sqrt {-a^{2}+b^{2}}-4 k \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right )}{\lambda \sqrt {-a^{2}+b^{2}}}}}{k +y}\right )\]

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