# Thread: Ordinary Differential Equation Problem

1. ## Ordinary Differential Equation Problem

Show that if Y (t); an n by n matrix, is the unique maximal solution of the
matrix IVP
Y'(t) = A*Y(t); Y (0) = I;
then Y has the characteristic property of exponentials, namely
Y (s + t) = Y (s)Y (t) for all s; t E(element of) R.
and
Y (-t) = Y(t)^-1

2. One way would be to use the fact that $\displaystyle Y(t) = e^{At}$, where $\displaystyle e^{At}$ is defined as the sum of the series $\displaystyle \sum_{n=0}^\infty \frac{A^nt^n}{n!}$ (of course, you have to show that the series converges in some suitable sense). Then the same proof which shows that $\displaystyle e^{s+t} = e^se^t$ can be used in this context.