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?

  • 1
    $\begingroup$ Try $cte^{2t}$. Integrating factors are fine too. $\endgroup$ – André Nicolas Mar 31 '16 at 17:00
  • $\begingroup$ @AndréNicolas Good suggestion! $\endgroup$ – 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 $c$.


You can always use integrating factors method (as suggested by Andre):

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.