SOLUTION: sopve the differential equation : (x^2-y^2)*dy=2xy*dx 1. dy*6x+dy(-y^2)=2x^2yd 2. 6dyx-dy^3-2x^2yd=0 3. d(6yx-y^3-2x^2y)=0 4. d=0

Algebra ->  Test -> SOLUTION: sopve the differential equation : (x^2-y^2)*dy=2xy*dx 1. dy*6x+dy(-y^2)=2x^2yd 2. 6dyx-dy^3-2x^2yd=0 3. d(6yx-y^3-2x^2y)=0 4. d=0       Log On


   

Question 964357: sopve the differential equation :
(x^2-y^2)*dy=2xy*dx
1. dy*6x+dy(-y^2)=2x^2yd
2. 6dyx-dy^3-2x^2yd=0
3. d(6yx-y^3-2x^2y)=0
4. d=0
Found 2 solutions by kekegeter , robertb:
Answer by kekegeter (97) About Me  (Show Source):
You can put this solution on YOUR website!
(x^2-y^2)*dy=2xy*dx
1. dy*6x+dy(-y^2)=2x^2yd
2. 6dyx-dy^3-2x^2yd=0
3. d(6yx-y^3-2x^2y)=0
4. d=0
Hope that this will help you!

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%5E2-y%5E2%29dy=2xydx+ is an example of an Euler-homogeneous DE of degree 2.
This can be solved by the substitution x = vy.
==> dx = vdy + ydv.
==> %28x%5E2-y%5E2%29dy=2xydx+ <==> %28v%5E2y%5E2-y%5E2%29dy=2%28vy%29y%28vdy%2Bydv%29+
==> %28v%5E2+-+1%29dy+=+2v%5E2dy%2B2vydv
<==> -2vydv+=+%281%2Bv%5E2%29dy
<==> %28%282v%29%2F%281%2Bv%5E2%29%29dv+=+-dy%2Fy
==> ln%281%2Bv%5E2%29+=+-lny+%2B+lnC
==> ln%28y%281%2Bv%5E2%29%29+=+lnC
==> y%281%2Bv%5E2%29+=+C
==> y%281%2Bx%5E2%2Fy%5E2%29+=+C
==> y%2Bx%5E2%2Fy+=+C
==> x%5E2+%2B+y%5E2+=+Cy, the general solution to the original DE.