In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = e^2x (a + bx)

It is given y = e^2x(a + bx) -------(1)

Now, differentiating both side w.r.t. x, we get,

y’ = 2e^2x(a + bx) + e^2x.b ------(2)

Now, let us multiply equation (1) with 2 and then subtracting it to equation (2), we get,

y’ – 2y = e^2x(2a +2bx + b) – e^2x(2a + 2bx)

⇒ y’ – 2y = be^2x ---------(3)

Now, again differentiating both sides w.r.t. x, we get,

y” – 2y’ = 2be^2x ------(4)

Dividing equation (4) by equation (3), we get,

⇒ y” – 2y’ = 2y’ – 4y

⇒ y” – 4y’ – 4y = 0

Therefore, the required differential equation is y” – 4y’ - 4y = 0.