Solve the differential equation $(x - y^{2}) d x + 2 x y d y = 0$ .

y^{2} + x l n a x = 0

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y^{2} - y l n a x = 0

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y^{2} - x l n a x = 0

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

y^{2} + y l n a x = 0

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Solution

The correct option is A $y^{2} + x l n a x = 0$
$(x - y^{2}) d x + 2 x y d y = 0 \Rightarrow 2 y \frac{d y}{d x} - \frac{y^{2}}{x} = - 1$
Substitute $v = y^{2} \Rightarrow d v = 2 y d y$
$\frac{d v}{d x} - \frac{v}{x} = - 1$ ...(1)
Here $P = - \frac{1}{x} \Rightarrow \int P d x = - \int \frac{1}{x} d x = - log x = log \frac{1}{x}$
$∴ I . F . = e^{log \frac{1}{x}} = \frac{1}{x}$
Multiplying (1) by $I . F .$ we get
$\frac{1}{x} \frac{d v}{d x} - \frac{v}{x^{2}} = - \frac{1}{x}$
Integrating both sides w.r.t $x$ we get
$\frac{v}{x} = - \int \frac{1}{x} d x + a = - log x + a \Rightarrow y^{2} = x log a x$

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