The solution of the differential equation dy/dx + 2y = e^x is-

y = e^x /3 + Ce^-2x

The given differential equation is dy/dx + 2y = e^x

This ifs of the form dy/dx +Py = Q

where P = 2, Q= e^x

I.F. = e^∫Pdx = e^∫2dx =e^2x 

The solution of the given differential equation is given by-

y(I.F.) = ∫(Q xI.F.)dx + C

⇒ye^2x = ∫(e^x.e^2x)dx + C 

⇒ ye^2x = ∫(e^3x)dx + C 

⇒ye^2x = e^3x/3 + C

so, y = e^x /3 + Ce^-2x