Question:
If $\mathrm{y}=\mathrm{y}(\mathrm{x})$ is the solution of the differential equation,
$x \frac{d y}{d x}+2 y=x^{2}$ satisfying
$y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal to:
Correct Option: , 3
Solution:
$\frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{2}{x}\right) \mathrm{y}=\mathrm{x}$
$\Rightarrow$ I.F. $=x^{2}$
$\therefore y^{2}=\frac{x^{4}}{4}+\frac{3}{4}($ As, $y(1)=1)$
$\therefore \quad y\left(x=\frac{1}{2}\right)=\frac{49}{16}$
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