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So I know how to find particular solutions for many systems of ODE, but I cannot solve a particular type using the method of undetermined coefficients because I do not know what ansatz to use.

Basically, in examples like the ones below:

$$\frac{d\vec{x}}{dt}=\left( {\begin{array}{cc} 4 & -2 \\ 8 & -4 \\ \end{array} } \right)\vec{x}+\left( {\begin{array}{cc} 1 \\ 0 \\ \end{array} } \right)t^{-3}+\left( {\begin{array}{cc} 0 \\ -1 \\ \end{array} } \right)t^{-2},\;\;\; t>0$$

the particular solution is $\vec{y}_p(t)=-\frac{1}{2}\left( {\begin{array}{cc} 1 \\ 0 \\ \end{array} } \right) t^{-2}+\left( {\begin{array}{cc} 2 \\ 5 \\ \end{array} } \right)t^{-1}-2\left( {\begin{array}{cc} 1 \\ 2 \\ \end{array} } \right)\ln(t)-2\left( {\begin{array}{cc} 1 \\ 2 \\ \end{array} } \right)$, but I do not know where $\ln(t)$ comes from. How do I find the ansatz to find the particular solution when negative powers are being used?

Thanks a lot!

s1047857
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  • If you want to avoid guessing from out of the blue and deduce the answer instead. See this. – Git Gud Aug 09 '14 at 19:19
  • Thanks! Though I have to say that's confused me even more :/... Isn't that link about the homogeneous equation, and not the non-homogeneous one? There should be a very simple rule for it because the problem is not worth many marks, and there are simple "rules" when it is an exponential, a cosine/sine, etc. I know how to solve it very quickly using variation of parameters, but I would like to know how do it using several methods just in case I get asked to use a specific one in the exam. Unless this case is not that easy with undetermined coefficients... – s1047857 Aug 09 '14 at 19:36

1 Answers1

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Don't try any ansatz, just solve. Namely, find two solutions $\vec{x}_1$, $\vec{x}_2$ of the homogeneous problem and form out of them the fundamental matrix $X_h$. Next looking for the solution of the inhomogeneous problem $\vec{x}'=A\vec{x}+\vec{f}$ in the form $X_h \vec{c}(t)$, one obtains an equation for $\vec{c}(t)$: $$X_h\vec{c}\,'=\vec{f}\quad \Longrightarrow\quad \vec{c}=\int^t X_h^{-1}(s)\vec{f}(s)\,ds.$$

In your case $\vec{x}_1=\left(\begin{array}{c}1+4t \\ 8t & \end{array}\right)$, $\vec{x}_2=\left(\begin{array}{cc} -2t \\ 1-4t\end{array}\right)$, hence $$X_h=\left(\begin{array}{cc}1+4t & -2t \\ 8t & 1-4t\end{array}\right)\quad \Longrightarrow \quad X_h^{-1}=\left(\begin{array}{cc}1-4t & 2t \\ -8t & 1+4t\end{array}\right),$$ and therefore the complete solution - say, for the non-homogeneity of the form $\left(\begin{array}{c}0 \\ -1 \end{array}\right)t^{-2}$ - will have the following form: \begin{align} \vec{x}(t)&=X_h(t)\int^t \left(\begin{array}{cc}1-4s & 2s \\ -8s & 1+4s\end{array}\right)\left(\begin{array}{c}0 \\ -1 \end{array}\right)s^{-2}ds=\\ &=X_h(t)\int^t\left(\begin{array}{c} -2s^{-1} \\ -s^{-2}-4s^{-1} \end{array}\right)ds=\\ &=\left(\begin{array}{cc}1+4t & -2t \\ 8t & 1-4t\end{array}\right)\left(\begin{array}{c} -2\ln t+c_1 \\ t^{-1}-4\ln t+c_2 \end{array}\right). \end{align}

Remark: If you absolutely want to do this exercise with undetermined coefficients, the approach is as follows. For constant $A$ one obtains the same powers of $t$ in $X^{−1}_h$ as in $X_h$ (here $1$ and $t$). By multiplying them by the nonhomogeneity of the form $t^{−2}$ and integrating the only terms that can arise are $t^{−1}$ and $\ln t$. Hence to find a particular solution you should try their linear combination. For the non-homogeneity of the form $t^{−3}$ you would similarly have to try a linear combination of $t^{−2}$ and $t^{−1}$.

Start wearing purple
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    just for the sake of completeness, let me add that this method is known as "variation of parameters" (for systems of ODEs). a quick search on google returned me the following file, which has a summary of the theory behind the technique and two solved examples: http://www.mth.msu.edu/~sen/math_235/lectures/lec_20s_var_par_sys.pdf – etothepitimesi Aug 09 '14 at 22:14
  • @el.Salvador Thanks, sure. My (maybe false) impression was that the OP knew about various methods and his main problem was their practical implementation. – Start wearing purple Aug 09 '14 at 22:17
  • Thanks a lot! We were not shown any examples of how to solve this type of ODE using undetermined coefficients (since we can use any method that we want), but it really bugged me that I couldn't solve it using this particular method. – s1047857 Aug 10 '14 at 22:53