If x^2+y^2=1 , prove that: y^3 d^2y/dx^2 +1=0

Question

If x^2+y^2=1 , prove that :

y^3 d^2y/dx^2 +1=0



I need the answer step by step.

Answer 1

Implicit defferentiation give us: 2x + 2ydy/dx = 0, dy/dx = - x/y.

Let differentiate again, we get: d²y/dx² = - (y - x·dy/dx)/y² = - (y + x²/y)/y² = - (y² + x²)y³ = - 1/y³.

So y³·d²/dx² + 1 = -1/y³·y³ + 1 = - 1 + 1 = 0