Define a function
for , and . Define a second function
Prove that the function satisfies the differential equation
Further, find all solutions to this differential equation on the interval. Prove that there is no solution such that the initial condition is satisfied. Why does this not contradict Theorem 8.3?
Proof. To prove that satisfies the given differential equation we use Theorem 8.3 to find a formula for all solutions to this differential equation. For we have,
Therefore, using Theorem 8.3 we let
Hence, is a solution satisfying . Also, since and are both 0 when , is valid on the entire interval .
Finally, the differential equation has no solution satisfying (since for all ). This does not contradict Theorem 8.3 since is not continuous on any interval around