If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2}+6y$ is equal to

$5\frac{dy}{dx}$

The correct answer is Option (1) → $5\frac{dy}{dx}$

Given: $y = 3e^{2x} + 2e^{3x}$

First derivative: $\frac{dy}{dx} = 6e^{2x} + 6e^{3x}$

Second derivative: $\frac{d^2y}{dx^2} = 12e^{2x} + 18e^{3x}$

Now compute: $\frac{d^2y}{dx^2} + 6y$

$= 12e^{2x} + 18e^{3x} + 6(3e^{2x} + 2e^{3x})$

$= 12e^{2x} + 18e^{3x} + 18e^{2x} + 12e^{3x}$

$= 30e^{2x} + 30e^{3x}$

$= 6(5e^{2x} + 5e^{3x})$

$= 5(6e^{2x} + 6e^{3x})$

$= 5 \cdot \frac{dy}{dx}$