Find the general solution of (x+2y^3) dy dx =y. - Vidyadip

Question

Find the general solution of $(\text{x}+2\text{y}^3)\frac{\text{dy}}{\text{dx}}=\text{y}.$

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Answer

We have, $(\text{x}+2\text{y}^3)\frac{\text{dy}}{\text{dx}}=\text{y}$
$\Rightarrow\text{y}.\frac{\text{dx}}{\text{dy}}=\text{x}+2\text{y}^3$
$\Rightarrow\frac{\text{dx}}{\text{dy}}=\frac{\text{x}}{\text{y}}+2\text{y}^2$ [Dividing both sides by y]
$\Rightarrow\frac{\text{dx}}{\text{dy}}-\frac{\text{x}}{\text{y}}=2\text{y}^2$
which is a linear differential equation.
On comparing it with $\frac{\text{dy}}{\text{dx}}+\text{P}\text{x}=\text{Q},$ we get
$\text{P}=-\frac{1}{\text{y}},\text{Q}=2\text{y}^2$
$\text{I.F}=\text{e}^{\int-\frac{1}{\text{y}}\text{dy}}$
$\text{I.F}=\text{e}^{-\int\frac{1}{\text{y}}\text{dy}}$
$\therefore\text{I.F.}=\text{e}^{-\log\text{y}}$
$\text{I.F.}=\frac{1}{\text{y}}$
The general solution is,
$\text{x}.\frac{1}{\text{y}}=\int2\text{y}^2.\frac{1}{\text{y}}\text{dy}+\text{C}$
$\Rightarrow\frac{\text{x}}{\text{y}}=\frac{2\text{y}^2}{2}+\text{C}$
$\Rightarrow\frac{\text{x}}{\text{y}}=\text{y}^2+\text{C}$
$\Rightarrow\text{x}=\text{y}^3+\text{Cy}$

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