# Examples of system of linear differential equations with periodic coefficients

 Math Emeritus Sci Advisor Thanks PF Gold P: 39,488 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)".