The differential equation whose solution is $y^{2} = 3 a y - x^{3}$ is

(x^{3} - y^{2}) \frac{d y}{d x} = 3 x^{2} y

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

(x^{3} - y^{2}) \frac{d y}{d x} = 3 x y

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(x^{3} - y) \frac{d y}{d x} = 3 x y^{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(y^{2} - x^{3}) \frac{d y}{d x} = 3 x y

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $(x^{3} - y^{2}) \frac{d y}{d x} = 3 x^{2} y$
$y^{2} = 3 a y - x^{3}$ ........... (1)

Differentiate with respect to $x$
$2 y y_{1} = 3 a y_{1} - 3 x^{2}$

$\frac{2 y y_{1} + 3 x^{2}}{y_{1}} = 3 a$

Put in (1)
$\Rightarrow y^{2} = \frac{y}{y_{1}} (2 y y_{1} + 3 x^{2}) - x^{3}$

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

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

but $y_{1} = \frac{d y}{d x}$

$(x^{3} - y^{2}) \frac{d y}{d x} = 3 x^{2} y$

Which is the required differential equation.

Suggest Corrections