Q:

Finding general solution of elementary differential equation

Can this equation be solved using extended linearity principle? 
$$ \frac{dy}{dt} - 2y = 7e^{2t} $$
I found the general solution to the homogeneous portion: $ y = ke^{2t} $. 
But finding a particular solution to the nonhomogeneous equation is difficult, as the terms keep cancelling out. 
Is this a candidate for the methodology of integrating factors? 

A:

When the inhomogeneous part of your ODE is of the form $Ae^{bt}$, a good guess for your particular solution is $ce^{bt}$, unless your homogeneous solution already has that as its form. In this case, the standard method is to multiply by $t$, and guess $cte^{bt}$ as your particular solution. If you guess, as Andre Nicolas suggests, $ct e^{2t}$ as your particular solution, you will not get complete cancellation, and you will be able to determine the value of $c$.

