Question:

<p><strong>Step 1:</strong> Given $y^2 = 3\cos^2 x + 2\sin^2 x = 2 + \cos^2 x$</p><p><strong>Step 2:</strong> Differentiate: $2y\frac{dy}{dx} = -2\cos x \sin x = -\sin(2x)$</p><p><strong>Step 3:</strong> So $y\frac{dy}{dx} = -\frac{1}{2}\sin(2x)$</p><p><strong>Step 4:</strong> Differentiate again: $y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = -\cos(2x)$</p><p><strong>Step 5:</strong> From step 2: $\left(\frac{dy}{dx}\right)^2 = \frac{\sin^2(2x)}{4y^2}$</p><p><strong>Step 6:</strong> Calculate $y^4 + y\frac{d^2y}{dx^2} = y^4 + (-\cos(2x) - \left(\frac{dy}{dx}\right)^2)$</p><p><strong>Step 7:</strong> Substitute and simplify using $y^2 = 2 + \cos^2 x$ to get the answer as 6.</p>