Given \( y = 5 \cos x - 3 \sin x \), the first derivative with respect to \( x \) is: \[ \frac{dy}{dx} = \frac{d}{dx} \left( 5 \cos x - 3 \sin x \right) \] Using standard derivative rules \( \frac{d}{dx} (\cos x) = -\sin x \) and \( \frac{d}{dx} (\sin x) = \cos x \): \[ \frac{dy}{dx} = -5 \sin x - 3 \cos x \] Differentiating \( \frac{dy}{dx} \) to find the second derivative, \( \frac{d^2y}{dx^2} \): \[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( -5 \sin x - 3 \cos x \right) \] Applying the derivative rules: \[ \frac{d^2y}{dx^2} = -5 \cos x + 3 \sin x \] Now, we compute the sum of \( \frac{d^2y}{dx^2} \) and \( y \): \[ \frac{d^2y}{dx^2} + y = \left( -5 \cos x + 3 \sin x \right) + \left( 5 \cos x - 3 \sin x \right) \] Combining like terms: \[ = (-5 \cos x + 5 \cos x) + (3 \sin x - 3 \sin x) \] \[ = 0 \] This confirms that: \[ \frac{d^2y}{dx^2} + y = 0 \]