Question

Find a differential equation corresponding to $y=a{x}^2+bx$

Answer 1

Let $y=a{x}^2+bx$.......(1).

Since we have two arbitrary constant in the given equation so we will be getting a second order differential equation.

Differentiating both sides with respect to $x$ we get,

$y^{\prime }=2ax+b$.......(2)

Again differentiating both sides of (2) w.r.to $x$ we get,

$y^{\prime \prime }=2a$

or, $a=\frac{y^{\prime \prime }}{2}$......(3).

Using (3) in (2) we get,

$b=(y^{\prime }-x{y}^{\prime \prime })$.

Using these values of $a$ and $b$ in equation (1) we get,

$y=\frac{y^{\prime \prime }}{2}{x}^2+(y^{\prime }-x{y}^{\prime \prime })x$.

This is the required differential equation.