Solve d y d x +2 x y=x^2 - Vidyadip

Question

Solve $\frac{d y}{d x}+\frac{2}{x} y=x^2$

✓

Answer

$
\frac{d y}{d x}+\frac{2}{x} y=x^2
$
This is a linear differential equation of the form
$
\begin{aligned}
& \frac{d y}{d x}+P y=Q, \text { where } P=\frac{2}{x}, Q \\
& \therefore \text { I.F. }=e^{\int P d x}=e^{2 \int \frac{1}{x} d x} \\
& =e^{2 \log x}=e^{\log x^2}=x^2
\end{aligned}
$
$\therefore$ the solution of (1) is given by
$
\begin{aligned}
y \cdot(\text { I.F. }) & =\int Q\left(\text { I.F.) } d x+c_1\right. \\
\therefore y x^2 & =\int x^2 \cdot x^2 d x+c_1 \\
& =\int x^4 d x+c_1
\end{aligned}
$
$
\begin{aligned}
& \therefore y x^2=\frac{x^5}{5}+c_1 \\
& \therefore 5 x^2 y=x^5+c \text {, where } c=5 c_1
\end{aligned}
$
This is the general solution.

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