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Math Help - general solution for differential equations

  1. #1
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    general solution for differential equations

    Hi I wasn't sure whether to post this in the linear algebra forum or the calculus forum, but it deals with differential equations. I'm having a hard time understanding this one question and how it differs from the normal case. The question is:

    Find the general real-valued solution for the system of equations:

    y'_1 = 3y_1 + y_2

    y'_2 = y_1 + 3y_2

    Now, I know how to solve equations of the form

    y' = 3y + f

    where f is just some function or constant. The confusing part for this problem is that y_1 & y_2 are in both equations, so you can't just solve each equation outright like you typically would when there is just y and not the y1 and y2.

    Once the general equation is known, I know its pretty easy to solve the initial value problem I need to solve after, but I just need help on how to approach the problem.

    Am I supposed to somehow substitute to get one equation with all y1 and the other with all y2 or how would I go about doing it? It just seems like a weird problem to me that I've never encountered before.
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  2. #2
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    We could write it as:

    x'=2x+y

    y'=x+3y

    Now, we can find the eigenvalues:

    \begin{bmatrix}3-{\lambda}&1\\1&3-{\lambda}\end{bmatrix}

    {\lambda}^{2}-6{\lambda}+8=0

    {\lambda}=2, \;\ {\lambda}=4

    Now, can you proceed?.

    Start by plugging the eigenvalues we just found back in for lambda in the matrix above.
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  3. #3
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    Yes, thanks that makes A LOT more sense lol, thank you very much.
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