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I've recently encountered the problem of solving $y'' - 2y' + y = te^t$ by the Method of Undetermined Coefficients.

The complementary solution is $y_c(t) = c_1e^t + c_2te^t$.

So I choose $y_p(t) = At^2e^t$ as a particular solution. However, this does not work and the solution says the appropriate particular solution is $y_p(t) = At^3e^t + Bt^2e^t$.

Is there a way to identify that my initial choice of $y_p$ would not work without having to do the necessary calculations?

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1 Answer 1

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For a right side term $$p(t)e^{rt}$$ the corresponding solution term is $$t^mq(t)e^{rt}$$ where $p$, $q$ are polynomials with $\deg p=\deg q$ and $m$ is the multiplicity of $r$ as root of the characteristic polynomial of the left side.

Here $p(t)=t$, thus $q(t)=At+B$, and with $m=2$ the stated form results.

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