Examples of system of linear differential equations with periodic coefficients

[princy]

hi ,
can anybody give me some examples of 'systems of linear differential equations with periodic coefficients'? i don't know how to solve it.. where can i get problems and solutions on this?

 

[HallsofIvy]

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

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:
\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}
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)".