I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$.
This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up and read the solution (which I quickly regretted, but at the same time realized that I had not carefully read one very important theorem* in the text) to find that it begins with:
We know $f''(t)=f'(t)$.
How do we know this? Also, in general, how would you have approached this problem? Any solution with your thoughts written out would be very appreciated. I am not interested in the solution, but the thought process behind this.
*For the curious, the theorem was that:
If $f$ is differentiable and $f'(x)=f(x)$ for all $x$ then there is a number $c$ s.t. $f(x)=ce^x$ for all $x$.