If `y^2=a^2cos^2x+b^2sin^2x` then prove that `(d^2y)/(dx^2)+y=(a^2b^2)/(y^3)`

Explanation

To prove that \(\frac{d^2y}{dx^2} + y = \frac{a^2b^2}{y^3}\) given that \(y^2 = a^2 \cos^2 x + b^2 \sin^2 x\), we will follow these steps:

### Step 1: Differentiate \(y^2\)
Starting with the equation \(y^2 = a^2 \cos^2 x + b^2 \sin^2 x\), we differentiate both sides with respect to \(x\):

\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(a^2 \cos^2 x + b^2 \sin^2 x) \]
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