Solution of differential equation $xdx + ydy = \sqrt{x^2 + x^2y^2}dx$ is Option 1) $2\sqrt{x^2 + y^2} -x^2 =c$ Option 2) $2\sqrt{x^2 + y^2} + x^2 =c$ Option 3) $2\sqrt{x^2 + y^2} -x=c$ Option 4) $2\sqrt{x^2 + y^2}+ x =c$

Answers (1)

As we have learnt,

General form of Variable Separation -

$d\left (\sqrt{x^{2}+y^{2}} \right )= \frac{xdx+ydy}{\sqrt{x^{2}+y^{2}}}$

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Given equation can be written as -

$xdx + ydy = x\sqrt{x^2 + y^2}dx \\*\Rightarrow \frac{xdx + ydy}{\sqrt{x^2 + y^2}} = xdx \\*\Rightarrow d\left(\sqrt{x^2 + y^2} \right ) = xdx$

On Integrating we get

$\sqrt{x^2 + y^2} =\frac{x^2}{2} + c \\*\Rightarrow 2\sqrt{x^2 + y^2} -x^2 = c$

Option 1)

$2\sqrt{x^2 + y^2} -x^2 =c$

Option 2)

$2\sqrt{x^2 + y^2} + x^2 =c$

Option 3)

$2\sqrt{x^2 + y^2} -x=c$

Option 4)

$2\sqrt{x^2 + y^2}+ x =c$

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