# The general solution of the differential equation

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)

1. (1) $y^{2}-2 x^{2}+c x^{3}=0$

2. (2) $y^{2}+2 x^{3}+c x^{2}=0$

3. (3) $y^{2}+2 x^{2}+c x^{3}=0$

4. (4) $y^{2}-2 x^{3}+c x^{2}=0$

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,$\left.c=2 c_{1}\right]$