Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.`y=e^(2x)(a+b x)`

Explanation

Given equation is

`y = e^{2x}(a+bx)` -(i)

Now, differentiate w.r.t x

`\frac{dy}{dx}= \frac{d(e^{2x}(a+bx))}{dx}= 2e^{2x}(a+bx)+e^{2x}.b` -(ii)

Now, again differentiate w.r.t x

`y^{''}= \frac{d^2y}{dx^2}= \frac{d}{dx}\frac{dy}{dx} = 4e^{2x}(a+bx)+2be^{2x}+2be^{2x}= 4e^{2x}(a+bx)+4be^{2x} -(iii)`

Now, multiply equation (ii) with 2 and subtract from equation
(iii)
`4e^{2x}(a+bx)+4be^{2x}-2\left ( 2e^{2x}(a+bx)+be^{2x} \right )=y^{''}-2y^{'}\\ \\ 2be^{2x} = y^{''}-2y^{'}\\ \\ be^{2x}= \frac{y^{''}-2y^{'}}{2}` -(iv)

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