$y=Ae^{2x}+Be^{-5x}$----(i)

Step 1:

$\large\frac{dy}{dx}$$=2Ae^{2x}-5Be^{-5x}$

$\qquad= 2Ae^{2x}+2Be^{-5x}-7Be^{-5x}$

$\qquad=2y-7Be^{-5x}$-----(ii)

From (ii) $7Be^{-5x}=2y-\large\frac{dy}{dx}$ -----(iii)

Step 2:

Differentiating (ii) again w.r.t x

$\large\frac{d^2y}{dx^2}$$=2 \large\frac{dy}{dx}$$+35 Be^{-5x}$-----(iv)

Step 3:

Substituting in (iv) from (iii)

$\large\frac{d^2y}{dx^2}$$=2 \large \frac{dy}{dx}$$+5 [2y- \large\frac{dy}{dx}]$

$\large\frac{d^2y}{dx^2}$$+3\large\frac{dy}{dx}$$-10y=0$ is the DE