Show that the DE is exact, then solve it. 2.a²y' + 4.xy = 3 sin x, y(27) = 0

Let us consider first order differential equation $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$

The differential equation is solved by using integrating factor method.

The integrating factor is obtained as  

$I.F.=e^{\int P\left(x\right) dx}$

Then the solution of differential equation  is 

$\left(y{e}^{\int P\left(x\right)dx}\right)=\int Q\left(x\right)e^{\int P\left(x\right) dx} +C$