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If y=e2x (ax + b), show that y2–4y1+4y = 0.

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 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

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