Solve the differential equation dy dx +2xy=y. - Vidyadip

Question

Solve the differential equation $\frac{\text{dy}}{\text{dx}}+2\text{xy}=\text{y}.$

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Answer

We have $\frac{\text{dy}}{\text{dx}}+2\text{xy}=\text{y}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}+2\text{xy}-\text{y}=0$
$\Rightarrow\frac{\text{dy}}{\text{dx}}+(2\text{x}-1)\text{y}=0$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}(1-2\text{x})\text{dx}$
$\Rightarrow\frac{\text{dy}}{\text{y}}=(1-2\text{x})\text{dx}$
Integrating both sides we get,
$\int\frac{\text{dy}}{\text{y}}=\int(1-2\text{x})\text{dx}$
$\Rightarrow\log\text{y}=\text{x}-\text{x}^2+\log\text{C}$
$\Rightarrow\log\text{y}-\log\text{C}=\text{x}-\text{x}^2$
$\Rightarrow\log\frac{\text{y}}{\text{C}}=\text{x}-\text{x}^2$
$\Rightarrow\frac{\text{y}}{\text{C}}=\text{e}^{\text{x}-\text{x}^2}$
$\Rightarrow\text{y}=\text{C}\text{e}^{\text{x}-\text{x}^2}$

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