Proof That Given Equation is Implicit Solution of Differential Eqn

[Color_of_Cyan]

"Show that the given equation is an implicit solution of the given differential eqn"

Homework Statement

Show that the given equation is an implicit solution of the given differential equation -

y2 - 1 - (2y + xy)(y-prime) = 0

y2 - 1 = (x + 2)2

Homework Equations

y² - 1 - (2y + xy)(y-prime) = 0

y² - 1 = (x + 2)²

The Attempt at a Solution

I probably went wrong here: Solve for y-prime: y² - 1 = (x + 2 )²

y² = (x + 2)² + 1

y = [ (x + 2)² + 1 ]^1/2

y-prime (using chain rule)= 1/ { 2 [ ( x + 2 )² + 1 ]^1/2 } * 2(x + 2)

Then I would substitute y² = (x + 2)² + 1 into y² - 1 - (2y + xy)(y-prime) = 0Am I on the right track? Because the problem as you can probably see may get a little messy unless I have to check my algebra better or something. Thank you!

 

[Dick]

You aren't using the 'implicit differentiation' trick. If y^2=(x+1)^2+1 then you can find y' without solving for y. Just take d/dx of both sides. d/dx(y^2)=2*y*dy/dx. Take it from there.

 

[Color_of_Cyan]

Thanks I got it now, sorry to get back to this a bit late.