Question

Find the differential equation if y = e^ 2x (a + bx) .

Answer 1

Concept:

   Differential equations are formed by eliminating arbitrary constants occur in the given equation. In the given equation, there are 2 constants. So we have to differentiate 2 times the given equation to elimnate them.

Given:

$y=e^{2x}{a+bx}$

$e^{-2x}\;y=a+bx$

Differentiate with respect to 'x'

$e^{-2x}\;\frac{dy}{dx}+y\;e^{-2x}(-2)=b$

$e^{-2x}(\frac{dy}{dx}-2y)=b$

Differentiate with respect to 'x' once again

$e^{-2x}(\frac{d^2y}{dx^2}-2\frac{dy}{dx})+(\frac{dy}{dx}-2y)e^{-2x}(-2)=0$

$e^{-2x}[\frac{d^2y}{dx^2}-2\frac{dy}{dx}-2\frac{dy}{dx}+4y]=0$

$e^{-2x}[\frac{d^2y}{dx^2}-4\frac{dy}{dx}+4y]=0$

$\frac{d^2y}{dx^2}-4\frac{dy}{dx}+4y=0$ $\:\:(\because\;e^{-2x}{\neq}0)$

$\implies\boxed{\bf\frac{d^2y}{dx^2}-4\frac{dy}{dx}+4y=0}$       is the required differential equation