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The general solution of the differential equation $\frac{d y}{d x}+\frac{2}{x} y=x^2$ is
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Verified Answer
The correct answer is:
$y=c x^{-2}+\frac{x^3}{5}$
$y=c x^{-2}+\frac{x^3}{5}$
Given differential equation is
$$
\frac{d y}{d x}+\frac{2}{x} \cdot y=x^2
$$
This is of the linear form.
$$
\begin{aligned}
& \therefore P=\frac{2}{x}, Q=x^2 \\
& \text { I.F }=e^{\int \frac{2}{x} d x}=e^{\log x^2}=x^2
\end{aligned}
$$
Solution is
$$
\begin{aligned}
& y \cdot x^2=\int x^2 \cdot x^2 d x+c=\frac{x^5}{5}+c \\
& y=\frac{x^3}{5}+c x^{-2}
\end{aligned}
$$
$$
\frac{d y}{d x}+\frac{2}{x} \cdot y=x^2
$$
This is of the linear form.
$$
\begin{aligned}
& \therefore P=\frac{2}{x}, Q=x^2 \\
& \text { I.F }=e^{\int \frac{2}{x} d x}=e^{\log x^2}=x^2
\end{aligned}
$$
Solution is
$$
\begin{aligned}
& y \cdot x^2=\int x^2 \cdot x^2 d x+c=\frac{x^5}{5}+c \\
& y=\frac{x^3}{5}+c x^{-2}
\end{aligned}
$$
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