6.2.16 6.2

6.2.16.1 [653] problem number 1

problem number 653

Added January 14, 2019.

Problem 2.6.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

Contains unresolved integral

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6.2.16.2 [654] problem number 2

problem number 654

Added January 14, 2019.

Problem 2.6.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.16.3 [655] problem number 3

problem number 655

Added January 14, 2019.

Problem 2.6.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.16.4 [656] problem number 4

problem number 656

Added January 14, 2019.

Problem 2.6.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {\lambda c_1}{a \lambda \cos ^k(x+\lambda K[6016])+1}-\int _1^x-\frac {a k \lambda ^2 c_1 \cos ^{k-1}(K[1]+\lambda K[6016]) \sin (K[1]+\lambda K[6016])}{\left (a \lambda \cos ^k(K[1]+\lambda K[6016])+1\right )^2}dK[1]\right )dK[6016]+\int _1^x\frac {a \lambda c_1 \cos ^k(\lambda y+K[1])}{a \lambda \cos ^k(\lambda y+K[1])+1}dK[1]+c_2\right \}\right \}\]

Maple ✓

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

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

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6.2.16.5 [657] problem number 5

problem number 657

Added January 14, 2019.

Problem 2.6.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+( y^2-a^2 + a *lambda*cos(lambda*x) + a^2*cos(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'));
 

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

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6.2.16.6 [658] problem number 6

problem number 658

Added January 14, 2019.

Problem 2.6.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Cos[lambda*x]*y^2 + lambda*Cos[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*cos(lambda*x)*y^2 + lambda*cos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\sqrt {\pi }\, \sin \left (\lambda x \right ) {\mathrm e}^{-\sin \left (\lambda x \right )^{2}} \left (y -\sin \left (\lambda x \right )\right ) \operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right )+2 \sqrt {-\sin \left (\lambda x \right )^{2}}}{2 \sqrt {\pi }\, \sin \left (\lambda x \right ) {\mathrm e}^{-\sin \left (\lambda x \right )^{2}} \left (y -\sin \left (\lambda x \right )\right ) \operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right )+4 \sqrt {-\sin \left (\lambda x \right )^{2}}+\left (-2 y \sin \left (\lambda x \right ) \sqrt {\pi }+2 \sqrt {\pi }\, \sin \left (\lambda x \right )^{2}\right ) {\mathrm e}^{-\sin \left (\lambda x \right )^{2}}}\right )\]

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6.2.16.7 [659] problem number 7

problem number 659

Added January 14, 2019.

Problem 2.6.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {2 \cos \left (\frac {\lambda x}{2}\right ) e^{\frac {2 a \cos ^2\left (\frac {\lambda x}{2}\right )}{\lambda }} \left (y \cos \left (\frac {\lambda x}{2}\right )-\sin \left (\frac {\lambda x}{2}\right )\right )}{e^{\frac {2 a \cos ^2\left (\frac {\lambda x}{2}\right )}{\lambda }} (y \cos (\lambda x)-\sin (\lambda x)+y) \int _1^xe^{-\frac {2 a \cos ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \sec ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]+4}\right )\right \}\right \}\]

Maple ✓

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

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

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6.2.16.8 [660] problem number 8

problem number 660

Added January 14, 2019.

Problem 2.6.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+ ((lambda+a*cos(lambda*x)^2)*y^2 + lambda - a + a*cos(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'));
 

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

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6.2.16.9 [661] problem number 9

problem number 661

Added January 14, 2019.

Problem 2.6.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n y^m+b x) w_x +\cos ^k(\lambda y) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

restart; 
pde :=  (a*x^n*y^m+b*x)*diff(w(x,y),x)+ cos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (a \left (n -1\right ) \int y^{m} \cos \left (\lambda y \right )^{-k} {\mathrm e}^{b \int \cos \left (\lambda y \right )^{-k}d y \left (n -1\right )}d y +x^{-n +1} {\mathrm e}^{b \int \cos \left (\lambda y \right )^{-k}d y \left (n -1\right )}\right )\]

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6.2.16.10 [662] problem number 10

problem number 662

Added January 14, 2019.

Problem 2.6.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

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

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6.2.16.11 [663] problem number 11

problem number 663

Added January 14, 2019.

Problem 2.6.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n +b x \cos ^m y) w_x + \cos ^k(\lambda y) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

restart; 
pde :=  (a*x^n+b*x*cos(y)^m)*diff(w(x,y),x)+cos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (a \left (n -1\right ) \int \cos \left (\lambda y \right )^{-k} {\mathrm e}^{b \int \cos \left (\lambda y \right )^{-k} \cos \left (y \right )^{m}d y \left (n -1\right )}d y +x^{-n +1} {\mathrm e}^{b \int \cos \left (\lambda y \right )^{-k} \cos \left (y \right )^{m}d y \left (n -1\right )}\right )\]

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6.2.16.12 [664] problem number 12

problem number 664

Added January 14, 2019.

Problem 2.6.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n \cos ^m y + b x) w_x + \cos ^k(\lambda y) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

restart; 
pde :=  (a*x^n*cos(y)^m+b*x)*diff(w(x,y),x)+cos(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'));
 

\[w \left (x , y\right ) = f_{1} \left (a \left (n -1\right ) \int \cos \left (y \right )^{m} \cos \left (\lambda y \right )^{-k} {\mathrm e}^{b \int \cos \left (\lambda y \right )^{-k}d y \left (n -1\right )}d y +x^{-n +1} {\mathrm e}^{b \int \cos \left (\lambda y \right )^{-k}d y \left (n -1\right )}\right )\]

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