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6.21 Example

Find the general solution to the equation

\frac{dy}{dx}+\frac{y}{x}=x^{2}.

In this case $p(x)=\frac{1}{x}$ , so the integrating factor

I(x)={e^{\int dx/x}=}{e^{\log x}=x.}

Multiplying through by ${x,}$ we obtain the equation:

x\frac{dy}{dx}+y={\frac{d}{dx}\left(xy\right)}=x^{3}

Integrating both sides and dividing by $x$ , we obtain the general solution $y={\frac{x^{3}}{4}+\frac{c}{x}}$ , where $c$ is a constant.