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Checking solution for a system of ODEs

  1. Apr 11, 2013 #1
    1. The problem statement, all variables and given/known data

    Find a particular solution to
    ##\begin{cases}x'=5x+4y+t\\ y'=x+8y-t\end{cases}##
    using a variety of methods that are listed. I've been using undetermined coefficients on this problem thus far.

    2. Relevant equations

    3. The attempt at a solution

    My answer is
    ##\begin{cases}

    x(t)= -4C_1e^{4t}+C_2e^{9t}-\frac{1}{3}t-\frac{5}{54}\\

    y(t)= C_1e^{4t}+C_2e^{9t}+\frac{1}{6}t+\frac{7}{216}

    \end{cases}##

    I'll show my work in another post. This problem just so happens to have its solution listed in the back of the book, which is
    ##\begin{cases}x(t)=\frac{1}{5}C_1\left(e^{9t}+e^{4t}\right) + \frac{4}{5}C_2\left(e^{9t}-e^{4t}\right) - \frac{18t+5}{54}\\
    y(t) = \frac{1}{5}C_1\left(e^{9t}-e^{4t}\right) + \frac{1}{5}C_2\left(e^{9t}+e^{4t}\right) - \frac{t}{6}+\frac{7}{216}\end{cases}##

    I've also checked with WolframAlpha, which gives me a similar result with powers of ##e## factored out, but without the rational coefficients:
    http://www.wolframalpha.com/input/?i=x'=5x+4y+t,+y'=x+8y-t

    I'm wondering why all these answers may be different. It could be that I'm missing something in my work. Any ideas?

    I just don't see how the exponential terms in my answer can be written in the other ways.
     
  2. jcsd
  3. Apr 11, 2013 #2
    Here's my work. Rewriting the equations in matrix form, I get
    ##\begin{pmatrix}x\\y\end{pmatrix}' = \begin{pmatrix}5&4\\1&8\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} + t\begin{pmatrix}1\\-1\end{pmatrix}##

    Using the usual eigenvalue method, I get a characteristic solution of
    ##\begin{pmatrix}x\\y\end{pmatrix}_c = C_1e^{4t}\begin{pmatrix}-4\\1\end{pmatrix} + C_2e^{9t}\begin{pmatrix}1\\1\end{pmatrix}##.

    Then, using the undetermined coefficients method, as a guess I use
    ##\begin{pmatrix}x\\y\end{pmatrix}_p=t\vec{a}+\vec{b}##,

    which gives me the particular solution
    ##\begin{pmatrix}x\\y\end{pmatrix}_p=t\begin{pmatrix}-\frac{1}{3}\\\frac{1}{6}\end{pmatrix} + \begin{pmatrix}-\frac{5}{54}\\\frac{7}{216}\end{pmatrix}##

    So, my final solution, as before, is
    ##\begin{cases}

    x(t)= -4C_1e^{4t}+C_2e^{9t}-\frac{1}{3}t-\frac{5}{54}\\

    y(t)= C_1e^{4t}+C_2e^{9t}+\frac{1}{6}t+\frac{7}{216}

    \end{cases}##
     
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