Find the differential equation for the family of curves given by $y = e^{2x}(a + bx)$ by eliminating the arbitrary constants $a$ and $b$.

(N/A) $y = e^{2x}(a + bx)$ ...........$(1)$
Differentiating both sides with respect to $x$,we get:
$y' = 2e^{2x}(a + bx) + e^{2x}(b)$
$y' = 2y + be^{2x}$ ...........$(2)$
Rearranging equation $(2)$ to isolate the term with $b$:
$y' - 2y = be^{2x}$ ...........$(3)$
Differentiating both sides of equation $(3)$ with respect to $x$:
$y'' - 2y' = b(2e^{2x})$
$y'' - 2y' = 2(be^{2x})$
Substitute $be^{2x} = y' - 2y$ from equation $(3)$ into the above equation:
$y'' - 2y' = 2(y' - 2y)$
$y'' - 2y' = 2y' - 4y$
$y'' - 4y' + 4y = 0$
This is the required differential equation.