Note: y2 represents second order derivative i.e.\(\frac{d^2y}{dx^2}\) and y1 = dy/dx
Given,
y = e2x (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 \(\frac{d^2y}{dx^2}\)
As, \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\)
So, lets first find dy/dx
∵ y = e2x (ax + b)
Using product rule to find dy/dx:
Again differentiating w.r.t x using product rule:
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