Mathematics - Differential Equation Question with Solution | TestHub

Given differential equation can be written as

$\frac{dy}{dx}+\frac{2y}{x}=x$, which is a linear differential equation.

If$=e^{\int \frac{2}{x}dx}=e^{2\mathrm{l}\mathrm{n}x}=x^2$

Solution is

$y.x^2=\int x.x^{2}dx$

$\Rightarrow y.x^2=\frac{x^4}{4}+c$

Since, given curve passes through$\left(1,1\right)$

$\Rightarrow c=\frac{3}{4}$

Hence,$y\left(x\right)=\frac{x^2}{4}+\frac{3}{4x^2}$.

So,$y\left(\frac{1}{2}\right)=\frac{49}{16}$.