0
$\begingroup$

I would like to solve the following non-homogeneous second order differential equation with constant coefficients with the method of undetermined coefficient but I have some problem with the particular solution, can someone help me? Thanks in advance!

$2y^{''}+3y^{'}+y=t^{-2}$

My problem is that I don't know how to treat the negative power. For example considering $t^{2}$, I can write $Y=At^2+Bt+C$. But in this case what I should do?

$\endgroup$
  • 1
    $\begingroup$ @Moo. Did you try it ? I am almost sure that there is a typo since the particular solution involve very special functions. $\endgroup$ – Claude Leibovici Dec 2 '18 at 5:19
  • $\begingroup$ Thanks for the comments, so with undetermined coefficient is not possible to solve? $\endgroup$ – FTAC Dec 2 '18 at 5:41
  • 1
    $\begingroup$ What makes you think this is possible? Undetermined coefficients, at least the version in most books, applies only to functions $g(t)$ of very special form. $\endgroup$ – David C. Ullrich Dec 2 '18 at 14:32
2
$\begingroup$

First, the characteristic polynomial

$$ 2r^2 + 3r + 1 = (2r+1)(r+1) = 0 \implies r = -1, -1/2 $$

So the fundamental solution is $$ y_h(t) = c_1 e^{-t} + c_2e^{-t/2} $$


Now, you probably won't be able to use undetermined coefficients to obtain the particular solution, but variation of parameters can be of use. Let

$$ y_p(t) = u(t)e^{-t} + v(t)e^{-t/2} $$

Then

\begin{align} e^{-t}u' + e^{-t/2}v' &= 0 \\ -e^{-t}u' - \frac12 e^{-t/2}v' &= \frac12 t^{-2} \end{align}

$$ \implies u' = -t^{-2}e^t , \quad v' = t^{-2}e^{t/2} $$

The integrals are non-elementary, but can be expressed in terms of incomplete Gamma functions

$\endgroup$
  • $\begingroup$ Thanks a lot, now it's clear! $\endgroup$ – FTAC Dec 2 '18 at 19:17

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.