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Help me understand! - Differential equations - JEE Main-3

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)
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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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