Question

Form the differential equation to the curves
(i) $y^2=m(n^2-x^2)$ where $m,n$ are arbitrary constants
(ii) $a{x}^2+b{y}^2=1$ where $a$ and $b$ are arbitrary constants.
(iii) $xy=a{e}^{-x}+b{e}^x$

Answer 1

$\left(i\right)y^2=m{n}^2-m{r}^2$

$xy\frac{dy}{dy}=-2mn$

$\frac{ydy/dx}{x}=-m$

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

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

$\left(ii\right)a{x}^2+b{y}^2=1$

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

$ax=-by\frac{dy}{dx}$

$\frac{y}{x}\left(\frac{dy}{dx}\right)=-\frac{a}{b}$

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

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

$\left(iii\right)xy=a{e}^{-x}+b{e}^x$

$y+x\frac{dy}{dx}=-a{e}^{-x}+b{e}^x$

$\frac{dy}{dx}+\frac{dy}{dx}+x\frac{d^{2}y}{d{x}^2}=a{e}^{-x}+b{e}^x$

$2\frac{dy}{dx}+x\frac{d^{2}y}{d{x}^2}=xy$