How to solve this differential equation using Laplace transform?

Question

$$\dfrac{dy(t)}{dt}+\dfrac{1}{2}y(t).t=\dfrac{1}{2} $$ My attempt : $$sY(s)-\dfrac{1}{2}Y'(s)=\dfrac{1}{2s} \implies Y(s)=\dfrac{1}{2s^{2}}+\dfrac{1}{2s}Y'(s) $$ Now taking inverse laplace transform .. $$ y(t)=\dfrac{1}{2}t.u(t)+\underbrace{\dfrac{1}{2} \displaystyle \int_{0}^{t}\left[ \displaystyle \int_{0}^{\infty} e^{st}.Y'(s)ds\right] \,dt} $$ This underbraced integral i'm not able to solve, and the result has erfi(x) on it . Please check my method and suggest something , thanks :)