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

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Solution

To Solve the differential equation

$y d x - (x + 2 y^{2}) d y = 0$

This can be written as

$y d x = (x + 2 y^{2}) d y$

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

$\frac{d x}{d y} = \frac{x}{y} + \frac{2 y^{2}}{y}$

$\frac{d x}{d y} = \frac{x}{y} + \frac{2 y^{2}}{y}$

This is the Linear Differential Equation in "y"

So , Its Integrating Factor =

$I F = e^{\int \frac{- 1}{y} d y}$

$I F = e^{- l n y} = e^{l n y^{- 1}}$

$I F = y^{- 1} = \frac{1}{y}$

Now , Its Complete Solution will be

$x . I F = \int I F . (2 y) d y$

$x . \frac{1}{y} = \int \frac{1}{y} \times (2 y) d y$

$\frac{x}{y} = \int 2 d y$

$\frac{x}{y} = \frac{2 y^{2}}{2} + C$

$\frac{x}{y} = y^{2} + C$

Where C is the Integration Constant

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