The differential equation representing the curve $y = e^{2x} (a + bx)$, where $a, b$ are arbitrary constants is

$\frac{d^2y}{dx^2}-4\frac{dy}{dx}+4y=0$

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

Given family of curves:

$y = e^{2x}(a + bx)$

This contains two arbitrary constants ⇒ the differential equation must be of order 2.

Differentiate once:

$y' = 2e^{2x}(a+bx) + e^{2x}b$

$y' = e^{2x}(2a + 2bx + b)$

Differentiating again:

$y'' = 2e^{2x}(2a + 2bx + b) + e^{2x}(2b)$

$y'' = e^{2x}(4a + 4bx + 4b + 2b)$

$y'' = e^{2x}(4a + 4bx + 6b)$

Now eliminate the constants $a$ and $b$ using:

$y = e^{2x}(a + bx)$

$y' = e^{2x}(2a + 2bx + b)$

$y'' = e^{2x}(4a + 4bx + 6b)$

Divide each by $e^{2x}$ and form the combination:

$y'' - 4y' + 4y = 0$

The required differential equation is:

$y'' - 4y' + 4y = 0$