I'm confused about equation 11 in the first image (the second is added for context). How did they determine the left half of the equation and where did the right half come from? It looks like the halves disappeared on both sides, and the d/dt was factored out from dy/dt, producing d/dt(u(t)y). However, that doesn't seem correct. Can someone explain what happened between 10 and 11?
https://i.stack.imgur.com/ddR5d.png https://i.stack.imgur.com/bhM88.png
$$ \frac {dy}{dt}+ 1/2 y =1/2 e^{t/3}$$ $$e^{t/2}\frac {dy}{dt}+ 1/2e^{t/2} y =1/2 e^{t/2}e^{t/3}$$
$$\frac {d}{dt}( ye^{t/2})= 1/2 e^{t/2}e^{t/3}$$ $$ye^{t/2}=\int 1/2 e^{5t/6}dt+C$$ $$y=e^{-t/2}\int 1/2 e^{5t/6}dt+Ce^{-t/2}$$ $$y=Ce^{-t/2}+3/5 e^{t/3}$$