In the oral exam for my PhD, my advisor asked me a question about a differential equation. I don’t recall the question, but I remember the interaction that followed.
I was stuck, and my advisor countered by saying “Let me ask you a harder question.” I was still stuck, and so he said “Let me ask you an even harder question.” Then I got it.
By “harder” he meant “more general.” He started with a concrete problem, then made it progressively more abstract until I recognized it. His follow-up questions were logically harder but psychologically easier.
This incident came to mind when I ran across an example in Lawrence Evans’ control theory course notes. He uses the example to illustrate what he calls an example of mathematical wisdom:
It is sometimes easier to solve a problem by embedding it within a larger class of problems and then solving the larger class all at once.
The problem is to evaluate the integral of the sinc function:
He does so by introducing the more general problem of evaluating the function
which reduces to the sinc integral when α = 0.
We can find the derivative of I(α) by differentiating under the integral sign and integrating by parts twice.
As α goes to infinity, I(α) goes to zero, and so C = π/2 and I(0) = π/2.
Incidentally, note that instead of computing an integral in order to solve a differential equation as one often does, we introduced a differential equation in order to compute an integral.