$\frac{dy}{dx}+2y=f\left(x\right)$ is a linear differential equation.

Integrating Factor $=e^{\int 2dx}=e^{2x}$

Solution of the above equation is

$y. e^{2x}=\int f\left(x\right).e^{2x}dx+C$

$y\left(x\right)=e^{-2x}{\int }_0^{x}f\left(x\right).e^{2x}dx+C{e}^{-2x}$

$y\left(0\right)=0\Rightarrow C=0$

$\Rightarrow y\left(x\right)=e^{-2x}{\int }_0^{x}f\left(x\right).e^{2x}dx$

$\therefore y\left(\frac{3}{2}\right)=e^{-3}\left[{\int }_0^1{e}^{2x}dx+{\int }_1^{3/2}0.dx\right]=\frac{e^{-3}}{2}\left[e^2-1\right]=\frac{e^2-1}{2e^3}$