Question

Form the differential equation corresponding to the function $y=c(x-c{)}^2,c$ is an arbitrary constant.

Answer 1

$y=c(x-c{)}^2$ ------(1)

$y^1=2c(x-c)$

$\frac{y}{y^1}=\frac{c(x-c{)}^2}{2c(x-c)}=\frac{x-c}{2}$

$c=x-\frac{2y}{y^1}$

Substituting c in equation (1)

$[x-\frac{2y}{y^1}]{[x-[x-\frac{2y}{y^1}]]}^2$

$y=[x-\frac{2y}{y^1}]\left[\frac{4y^2}{(y^1{)}^2}\right]$

$y(y^1{)}^3=x^2{y}^1(y^1-1)+2y(x-2y)$