6.2.19 6.5

6.2.19.1 [692] problem number 1

problem number 692

Added January 20, 2019.

Problem 2.6.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

Has unresolved integrals

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6.2.19.2 [693] problem number 2

problem number 693

Added January 20, 2019.

Problem 2.6.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2-y \tan x+a(1-a) \cot ^2 x \right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\left (-\sin ^2(x)\right )^{\frac {1}{2} i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4}} \left (i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4} \cos (x)+2 y \sin (x)+\cos (x)\right )}{-i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4} \cos (x)+2 y \sin (x)+\cos (x)}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {\sin \left (x \right )^{2 a} \left (y +\cot \left (x \right ) a \right )}{\left (a -1\right ) \cos \left (x \right )-y \sin \left (x \right )}\right )\]

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6.2.19.3 [694] problem number 3

problem number 694

Added January 20, 2019.

Problem 2.6.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2-m y \tan x+b^2 \cos ^{2 m} x \right ) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=   diff(w(x,y),x)+ (y^2-m*y*tan(x)+b^2*cos(x)^(2*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 {3 \sqrt {\cos \left (x \right )^{2 m}}\, b \left (-\frac {\left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right ) \sin \left (x \right )^{2}}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \cos \left (x \right ) \cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+3 y \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \cos \left (x \right )^{m}}{3 \sqrt {\cos \left (x \right )^{2 m} \sec \left (x \right )^{2}}\, b \cos \left (x \right )^{2} \left (-\frac {\left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right ) \sin \left (x \right )^{2}}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sin \left (b \cos \left (x \right )^{-m +1} \sin \left (x \right ) \sqrt {\cos \left (x \right )^{2 m} \sec \left (x \right )^{2}}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )-3 y \cos \left (b \cos \left (x \right )^{-m +1} \sin \left (x \right ) \sqrt {\cos \left (x \right )^{2 m} \sec \left (x \right )^{2}}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \cos \left (x \right )^{m}}\right )\]

Mathematica answer is simpler

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6.2.19.4 [695] problem number 4

problem number 695

Added January 20, 2019.

Problem 2.6.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + m*y*Cot[x] + b^2*Sin[x]^m)*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+m*y*cot(x)+b^2*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'));
 

sol=()

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6.2.19.5 [696] problem number 5

problem number 696

Added January 20, 2019.

Problem 2.6.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 - 2*lambda^2*Tan[lambda*x]^2 - 2*lambda^2*Cot[lambda*x]^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^2-2*lambda^2*tan(lambda*x)^2-2*lambda^2*cot(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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 \lambda \cos \left (\lambda x \right )^{2}-y \sin \left (\lambda x \right ) \cos \left (\lambda x \right )-\lambda }{\left (2 \lambda \cos \left (\lambda x \right )^{2}-y \sin \left (\lambda x \right ) \cos \left (\lambda x \right )-\lambda \right ) \ln \left (\cos \left (\lambda x \right )+i \sin \left (\lambda x \right )\right )+2 i \sin \left (\lambda x \right ) \left (y \cos \left (\lambda x \right )^{3} \sin \left (\lambda x \right )+2 \lambda \cos \left (\lambda x \right )^{4}-\frac {y \sin \left (\lambda x \right ) \cos \left (\lambda x \right )}{2}-2 \lambda \cos \left (\lambda x \right )^{2}-\frac {\lambda }{2}\right ) \cos \left (\lambda x \right )}\right )\]

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6.2.19.6 [697] problem number 6

problem number 697

Added January 20, 2019.

Problem 2.6.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*(a + b) + 2*a*b + a*(lambda - a)*Tan[lambda*x]^2 + b*(lambda - b)*Cot[lambda*x]^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+lambda*(a+b)+2*a*b+a*(lambda -a)*tan(lambda*x)^2+ b*(lambda -b)*cot(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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (a \sin \left (\lambda x \right )^{2}-b \cos \left (\lambda x \right )^{2}-\cos \left (\lambda x \right ) y \sin \left (\lambda x \right )\right ) \left (a -\frac {3 \lambda }{2}\right ) \sin \left (\lambda x \right )^{\frac {2 b -\lambda }{\lambda }} \cos \left (\lambda x \right )^{\frac {2 a -\lambda }{\lambda }}}{-2 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {-a -b +2 \lambda }{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\left (\left (-3 \lambda ^{2}+\frac {\left (7 a +3 b \right ) \lambda }{2}-a b \right ) \cos \left (\lambda x \right )^{2}+y \sin \left (\lambda x \right ) \left (a -\frac {3 \lambda }{2}\right ) \cos \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}-\frac {5 \lambda \left (a -\frac {3 \lambda }{5}\right )}{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-a -b +\lambda }{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )}\right )\]

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6.2.19.7 [698] problem number 7

problem number 698

Added January 20, 2019.

Problem 2.6.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Cos[lambda*x]^n*y - a*Cos[lambda*x]^(n - 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*sin(lambda*x)* y^2 + a*cos(lambda*x)^n*y-a*cos(lambda*x)^(n-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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6.2.19.8 [699] problem number 8

problem number 699

Added January 20, 2019.

Problem 2.6.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-y \cos \left (\lambda x \right )+1}{\operatorname {Ei}_{1}\left (\frac {a \cos \left (\lambda x \right )}{\lambda }\right ) a \left (y \cos \left (\lambda x \right )-1\right )-{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{\lambda }} y \lambda }\right )\]

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6.2.19.9 [700] problem number 9

problem number 700

Added January 20, 2019.

Problem 2.6.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-y \cos \left (\lambda x \right )+1}{\operatorname {Ei}_{1}\left (\frac {a \cos \left (\lambda x \right )}{\lambda }\right ) a \left (y \cos \left (\lambda x \right )-1\right )-{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{\lambda }} y \lambda }\right )\]

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6.2.19.10 [701] problem number 10

problem number 701

Added January 20, 2019.

Problem 2.6.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( A e^{\lambda x} \cos (a y) + B e^{\mu x} \sin (a y) + A e^{\lambda x} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (A*Exp[lambda*x]*Cos[a*y] + B*Exp[mu*x]*Sin[a*y] + A*Exp[lambda*x])*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*exp(lambda*x)*cos(a*y) + B*exp(mu*x)*sin(a*y) + A*exp(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\tan \left (\frac {a y}{2}\right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{\mu x} B}{\mu }}+A \int {\mathrm e}^{\frac {-a \,{\mathrm e}^{\mu x} B +\lambda x \mu }{\mu }}d x a}{a \left (-\lambda +\mu \right )}\right )\]

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6.2.19.11 [702] problem number 11

problem number 702

Added January 20, 2019.

Problem 2.6.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \sin ^{n+1}(2 x) w_x + \left ( a y^2 \sin ^{2 n}x + b \cos ^{2 n} x \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-\frac {\cos \left (x \right )^{-\sqrt {n^{2}-4^{-n} a b}} \left (a y \sin \left (x \right )^{2 n}+\sin \left (2 x \right )^{n} \sqrt {n^{2}-4^{-n} a b}+\sin \left (2 x \right )^{n} n \right ) \sin \left (x \right )^{\sqrt {n^{2}-4^{-n} a b}}}{a y \sin \left (x \right )^{2 n}-\sin \left (2 x \right )^{n} \sqrt {n^{2}-4^{-n} a b}+\sin \left (2 x \right )^{n} n}\right )\]

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