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!


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?

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    $\begingroup$ @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
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    $\begingroup$ 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

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

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


\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

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

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