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Examples of system of linear differential equations with periodic coefficients

  1. Jul 16, 2011 #1
    hi ,
    can anybody give me some examples of 'systems of linear differential equations with periodic coefficients'? i dont know how to solve it.. where can i get problems and solutions on this?
  2. jcsd
  3. Jul 16, 2011 #2


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    Science Advisor

    You mean something like
    [tex]sin(t)\frac{dx}{dt}+ (1- t^2)\frac{dy}{dt}= e^t[/tex]
    [tex]cos(t)\frac{dx}{dt}+ t\frac{dy}{dt}= t[/tex]?

    You will want to try to reduce this to a single equation in either x only or y only.
    Essentially, use "Gaussian reduction" just as you would for an algebraic system.

    Or you could try writing the system as a matrix equation:
    [tex]\begin{bmatrix}sin(t) & 1- t^2 \\ cos(t) & t\end{bmatrix}\begin{bmatrix}\frac{dx}{dt} \\ \frac{dy}{dt}\end{bmatrix}= \begin{bmatrix}e^t \\ t\end{bmatrix}[/tex]
    and use the same matrix methods you would for the "constant coefficient" case. Of course, you would have to remember that, since the coefficient matrix now depends on t, d(AX)/dt= X(dA/dt)+ A(dX/dt), not just "A(dX/dt)".
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