Question

Find the differential equation of the family of curve represented by $c(y+c{)}^2+x^3$=0

Answer 1

Solution :

Given curve, $c(y+c{)}^2+x^3=0$___(1)

Differentiating w.r.t $x$,

$2c(y+c)\frac{dy}{dx}+3x^2=0$

From $e{q}^n\left(1\right)$

$c(y+c)=\frac{-x^3}{(y+c)}$

$\therefore \frac{-2x^3}{y+c}\frac{dy}{dx}+3x^2=0$

$\Rightarrow 3=\frac{2x}{y+c}\frac{dy}{dx}$

$\Rightarrow \frac{2x}{3}\frac{dy}{dx}=y+c$

substituting $c$ to equation $\left(1\right)$

$(\frac{2x}{3}\frac{dy}{dx}-y)(\frac{2x}{3}\frac{dy}{dx}{)}^2+x^3=0$

$\frac{4x^2}{9}\left(\frac{dy}{dx}{)}^2\right[\frac{2x}{3}\frac{dy}{dx}-y]+x^3=0$

$\Rightarrow \frac{4x^2}{9}{\left(\frac{dy}{dx}\right)}^3\times \frac{2x}{3}-y\times \frac{4x^2}{9}{\left(\frac{dy}{dx}\right)}^2+x^3=0$

$\Rightarrow {\left(\frac{2x}{3}\right(\frac{dy}{dx}{)}^3-\frac{4x^{2}y}{9}\left(\frac{dy}{dx}\right))}^2+x^3=0$