The equation of family of curves is
y = e^2x (a + b x)    ...(1)
Differentiating both sides w.r.t. x, we get,
y₁ = e^2x  b + (a + b x). 2 e^2x
or    y₁ = b e^2x + 2 y    [∵ of(1)]
∴ y₁ – 2 y = be^2x        ...(2)
Again differentiating w.r.t. x, we get,
y₂ – 2 y₁ = 2 b e^2x
or    y₂ – 2 y₁ = 2 (y₁ – 2 y)    [∵ of (2)]
or    y₂ – 2 y₁ = 2 y₁ – 4 y
or    y₂ – 4 y₁ + 4 y = 0, which is required differential equation.