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

.: We have, $y = {e^{2x}}\left( {a + bx} \right)$

…(1)
Differentiating (1) w.r.t. $x$,

we get
${y_1} = {e^{2x}}(b) + 2(a + bx){e^{2x}} \Rightarrow {y_1} = b{e^{2x}} + 2y$

(using (1))
$\Rightarrow {y_1} - 2y = b{e^{2x}}$

…(2)
Again differentiating (2) w.r.t. $x$,

we get
${y_2} - 2{y_1} = 2b{e^{2x}} \Rightarrow {y_2} - 2{y_1} = 2({y_1} - 2y)$

(using (2))
$\Rightarrow {y_2} - 4{y_1} + 4y = 0$,

which is the required differential equation.