Question

Construct a differential equation by eliminating the arbitrary constants $A,B,C$ for the equation $y^2=A{x}^2+Bx+C$.

Answer 1

Given the equation,

$y^2=A{x}^2+Bx+C$

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

$2y\frac{dy}{dx}=2Ax+B$

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

$2y\frac{d^{2}y}{d{x}^2}+2{\left(\frac{dy}{dx}\right)}^2=2A$

or, $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=A$

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

$y\frac{d^{3}y}{d{x}^3}+\frac{dy}{dx}\frac{d^{2}y}{d{x}^2}+2\left(\frac{dy}{dx}\right).\frac{d^{2}y}{d{x}^2}=0$

or, $y\frac{d^{3}y}{d{x}^3}+3\left(\frac{dy}{dx}\right).\frac{d^{2}y}{d{x}^2}=0$.

This is the required differential equation.