300 CHAPTER 12. SERIES AND TRANSFORMS

that

Φ(−t)x(t)−x0 =∫ t

0Φ(−s) f(s)ds,

x(t) = Φ(t)x0 +Φ(t)∫ t

0Φ(−s) f(s)ds

x(t) = Φ(t)x0 +∫ t

0Φ(t− s) f(s)ds

This is the variation of constants formula for the unique solution to the initial valueproblem. This has shown that if there is a solution, then it is the above. Next verifythat the above does solve the initial value problem applying fundamental theorem ofcalculus to the entries of the matrices. This completes somewhat more than what isaccomplished in an entire undergraduate differential equations course. Furthermore,unlike what is done in these wretched busy work courses, this leads somewhere.