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

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\frac{dy}{dx}+2xy=y

To find: Solution of the given differential equation

\int \frac{1}{x}{dx}=ln x+c\\ \\ \int x^{n}dx=\frac{x^{n+1}}{n+1}+c

Rewriting the given equation as,

\frac{dy}{dx}=y\left ( 1-2x \right )\\ \\ \frac{dy}{y}=\left ( 1-2x \right )dx

Integrate on both the sides,

\int \frac{dy}{y}=\int \left ( 1-2x \right )dx\\ \ln y =\left ( x-x^{2} \right )+\log c\\ \ln y-\ln c=x-x^{2}\\ \ln \frac{y}{c}=x-x^{2}\\ \frac{y}{c}=e^{x-x^{2}}\\ y=ce^{x-x^{2}}\\

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