Solve the following differential equation:
with when .
From Theorem 8.3 (page 310 of Apostol) we know that a differential equation of the form
on an interval has solution given by
In this particular case we apply the theorem (noting that in this problem we have is a function of , rather than a function of as in the theorem… of course, the names of the variables doesn’t matter, but we take care to apply the theorem properly) with
This gives us