Question

Construct a differential equation by eliminating the arbitrary constants $a$ and $b$ from the equation $a{x}^2+b{y}^2=1$.

Answer 1

We have the equation $a{x}^2+b{y}^2=1$ which contains two arbitrary constants so we will be having a second order differential equation.

Now differentiating with respect to $x$ both sides of the given equation we get,

$2ax+2by\frac{dy}{dx}=0$

or, $\frac{y}{x}\frac{dy}{dx}=-\frac{a}{b}$

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

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

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

This is the required differential equation.