Question

If $y=y\left(x\right)$ is the solution of the differential equation,
$x\frac{dy}{dx}+2y=x^2$ satisfying $y\left(1\right)=1$, then $y\left(\frac{1}{2}\right)$ is equal to

• A. $\frac{7}{64}$
• B. $\frac{13}{16}$
• C. $\frac{49}{16}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{49}{16}$

$\frac{dy}{dx}+\left(\frac{2}{x}\right)y=x$
$\Rightarrow I.F.=x^2$
$\therefore y{x}^2=\frac{x^4}{4}+\frac{3}{4}$ (As, $y\left(1\right)=1)$
$\therefore y(x=\frac{1}{2})=\frac{49}{16}$.