1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

First-Order Linear Differential Problem

  1. May 31, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve the following IVP:

    [itex]X' = \begin{pmatrix}2 & -1\\3 & -2\end{pmatrix}X + \begin{pmatrix}0\\t\end{pmatrix}[/itex] with [itex]X(0) = \begin{pmatrix}1\\0\end{pmatrix}[/itex]

    2. Relevant equations



    3. The attempt at a solution

    The eigenvalue corresponding to [itex]\begin{pmatrix}2 & -1\\3 & -2\end{pmatrix}[/itex] is [itex]\lambda = 0[/itex]. We find that [itex]X_c = c_1\begin{pmatrix}1\\2\end{pmatrix} e^{0t}[/itex]. Now in order to find [itex]X_p[/itex], how exactly is the right way? I took [itex]X_p = \begin{pmatrix}a_1\\b_1\end{pmatrix}t[/itex] and wanted to find [itex]a_1[/itex] and [itex]b_1[/itex]. Right or wrong?
     
  2. jcsd
  3. May 31, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Hiche! :smile:
    Nooo :redface:
     
  4. May 31, 2012 #3
    oh, crap! Apparently, 3 * 1 = 4 -_-

    So, again, the eigenvalues are [itex]\lambda_1 = -1[/itex] and [itex]\lambda_2 = 1[/itex]. I hope this is correct. So the solution of [itex]X_c = c_1\begin{pmatrix}1\\1\end{pmatrix}e^t + c_2\begin{pmatrix}1\\3\end{pmatrix}e^{-t}[/itex].

    Now about [itex]X_p[/itex]. Is my method correct (first post)?
     
  5. May 31, 2012 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    (i'm not sure, but…) i'd be inclined to go for [itex]X_p = \begin{pmatrix}a_0\\b_0\end{pmatrix} + \begin{pmatrix}a_1\\b_1\end{pmatrix}t[/itex]
     
  6. May 31, 2012 #5
    Okay, so upon a little work, [itex]X_p = \begin{pmatrix}1\\2\end{pmatrix}t[/itex] and the general solution is [itex]X = X_c + X_p[/itex]

    Thank you, tiny-tim.
     
  7. Jun 1, 2012 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    But [itex]X_p = \begin{pmatrix}1\\2\end{pmatrix}t[/itex] isn't a solution …

    Xp' = (1,2), and Xp(0) = (0,0)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook