Question

Construct a differential equation by eliminating the arbitrary constants $a,b$ from the equation:

$y=(ax+b)e^x$.

Answer 1

Given the equation,

$y=(ax+b)e^x$......(1)

Now differentiating both sides with respect to $x$ we get,

$\frac{dy}{dx}=(ax+b)e^x+a{e}^x$

or, $a{e}^x=\frac{dy}{dx}-y$. [ Using (1)]......(2)

Again differentiating both sides with respect to $x$ we get,

$\frac{d^{2}y}{d{x}^2}=(ax+b)e^x+2a{e}^x$

or, $\frac{d^{2}y}{d{x}^2}=2(\frac{dy}{dx}-y)+y$ [ Using (1) and (2)]

or, $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+y=0$.

This is the required differential equation.