# 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?

• Try $cte^{2t}$. Integrating factors are fine too. – André Nicolas Mar 31 '16 at 17:00
• @AndréNicolas Good suggestion! – d0rmLife Mar 31 '16 at 17:05

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 $c$.
Using Integrating factor: $e^{-2t}$ $$\frac{d}{dt}ye^{-2t} = 7$$
$$ye^{-2t} = \int 7 \ dt = 7t + C$$
$$y = e^{2t}(7t + C)$$
where $C$ is constant of integration