The general solution of the differential equation y 2 - x 3 d x - x y d y = 0 ,   x ≠ 0 is (where…

The general solution of the differential equation

(y^{2} - x^{3}) d x - x y d y = 0, (x \neq 0)

is (where c is a constant of integration)

$y^{2} + 2 x^{2} + c x^{3} = 0$
$y^{2} - 2 x^{2} + c x^{3} = 0$
$y^{2} - 2 x^{3} + c x^{2} = 0$
$y^{2} + 2 x^{3} + c x^{2} = 0$

Solution

Given differential equation can be written as:
$y^{2} d x - x y d y = x^{3} d x$
$\Rightarrow (y d x - x d y) = \frac{x^{3}}{y} d x$
$\Rightarrow \frac{y d x - x d y}{y^{2}} = {(\frac{x}{y})}^{3} d x$
$\Rightarrow \int \frac{d (\frac{x}{y})}{{(\frac{x}{y})}^{3}} = \int d x$
$\Rightarrow \frac{{(\frac{x}{y})}^{- 3 + 1}}{- 3 + 1} = x + c$

$\Rightarrow y^{2} = - 2 x^{3} - c x^{2}$

$\Rightarrow y^{2} + 2 x^{3} + c x^{2} = 0$

Asked in: JEE Main 2019 (12 Apr Shift 2)

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