Step 1:

$y= Ae^{2x} \cos (3x+B)$-----(i)$

$\large\frac{dy}{dx}$$=2 Ae^{2x} \cos (3x+B)-Ae^{2x}. 3 \sin (3x+B)$

(ie) $\large\frac{dy}{dx}$$=2y-3Ae^{2x} \sin (3x+B) $ -----(ii)

Step 2:

From (ii) $3Ae^{2x} \sin (3x+B)=2y- \large\frac{dy}{dx}$-----(iii)

Step 3:

Differentiate (ii) again

(ie) $\large\frac{d^2y}{dx^2}$$=2\large\frac{dy}{dx}$$-6Ae^{2x} \sin (3x+B)-3 Ae^{2x}, 3 \cos (3x+3)$-(ii)

(ie) $\large\frac{d^2y}{dx^2}$$=2\large\frac{dy}{dx}$$-2(2y -\large\frac{dy}{dx})$$-9y$ from (iii)

$\large\frac{d^2y}{dx^2}$$-4\large\frac{dy}{dx}$$+13y=0$ is the required D.E