If y=e^2x(ax + b), show that y₂–4y₁+4y = 0.

Note: y₂ represents second order derivative i.e. and y₁ = dy/dx

Given,

y = e^2x(ax + b) ……equation 1

to prove: y₂–4y₁+4y = 0

We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find

As,

So, lets first find dy/dx

∵ y = e^2x(ax + b)

Using product rule to find dy/dx:

……..equation 2

Again differentiating w.r.t x using product rule:

…….equation 3

In order to prove the expression try to get the required form:

Subtracting 4*equation 2 from equation 3:

Using equation 1:

∴ y₂–4y₁+4y = 0 ……..proved