examples of system of linear differential equations with periodic coefficients

by princy
Tags: coefficients, differential, equations, examples, linear, periodic
princy is offline
Jul16-11, 01:13 AM
P: 14
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?
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HallsofIvy is offline
Jul16-11, 07:22 AM
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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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