Explicit Solutions to this equation

vipertongn

1. The problem statement, all variables and given/known data
explicit solution of -2x^2y+y^2=1; 2xydx+(x^2-y)dy=0?


3. The attempt at a solution
I was able to find that dy/dx=-2xy/(x^2-y) which is the implicit solution of the equation. I pretty much derived both of the equations to make sure they are both equal to one another.... Not sure if that's how I could prove that the first order DE has them as an implicit solution...I'm just confused entirely... Now I'm trying to get the explicit solution...but that's just very hard considering the variables are stuck to each other.

Mentallic

How about after finding the derivative of both, set them equal and simplify? Notice that [itex]y\neq 0[/itex].

vipertongn

deriving both gives the same answer. Pretty much showing that one of them is an implicit solution. so y'= -2xy/(x^2-y) for both of them. haha in my first post I indicated what you said for me to do there

Mentallic

Oh sorry I forgot to mention that you should find an explicit solution of y(x) by realizing that it is a quadratic in terms of y.

vipertongn

ya, every time i try to get y by itself it ends up always not working out like y is always not isolated...

ehild

It is a quadratic equation in y!!!

y²-(2x²)y-1=0

Pretend that 2x² is some number b.

ehild