Question:
The general solution of the differential equation $\left(y^{2}-x^{3}\right) \mathrm{dx}-x y d y=0(x \uparrow 0)$ is :
(where $c$ is a constant of integration)
Correct Option: , 2
Solution:
Given differential equation can be written as,
$y^{2} d x-x y d y=x^{3} d x$
$\Rightarrow \frac{(y d x-x d y) y}{x^{2}}=x d x \Rightarrow-y d\left(\frac{y}{x}\right)=x d x$
$\Rightarrow-\frac{y}{x} \cdot d\left(\frac{y}{x}\right)=d x \quad \Rightarrow-\frac{1}{2}\left(\frac{y}{x}\right)^{2}=x+c_{1}$
$\Rightarrow 2 x^{3}+c x^{2}+y^{2}=0 \quad\left[\right.$ Here,$\left.c=2 c_{1}\right]$