If y = e^2xcos x , then which of the following is true?

The correct answer is Option (3) → \(\displaystyle\frac{d^{2}y}{dx^{2}} - 4\frac{dy}{dx} + 5y = 0\)

By the product rule:

    \( \displaystyle \frac{dy}{dx} = 2e^{2x}\cos x - e^{2x}\sin x = e^{2x}(2\cos x - \sin x) \)

Differentiating again for the second derivative:

    \( \displaystyle \frac{d^{2}y}{dx^{2}} = 2e^{2x}(2\cos x - \sin x) + e^{2x}(-2\sin x - \cos x) \)

    \( = e^{2x}(3\cos x - 4\sin x) \)

Now form the combination in Option (3):

    \( \displaystyle \frac{d^{2}y}{dx^{2}} - 4\frac{dy}{dx} + 5y = e^{2x}\left[(3\cos x - 4\sin x) - (8\cos x - 4\sin x) + 5\cos x\right] \)

    \( = e^{2x} \times 0 = 0 \)

Each of the other combinations leaves a non-zero multiple of \(e^{2x}\cos x\) or \(e^{2x}\sin x\), so only Option (3) holds for all x.