There is an important type of problem that is reoccuring when solving certain PDE's it is given below:
Now there are trivial solutions (uninteresting, i.e. ) and non-trivial solutions i.e. the "non-obvious".
Those value of are called the eigenvalues and those functions corresponding to that are called the eigenfunctions.
Remark 1: The book uses instead but I will not because it is more difficult in LaTeX.
Remark 2: They are called eigenvalues because they are similar to eigenvalues of a matrix. Remember the eigenvalues of a matrix is when it has non-trivial solutions , but that is something else. I am just saying the idea is the same.
It is extremely important for use to answer the following question.
"What are the eigenvalues and the eigenfunctions of the differencial equation ?"
There is a general theory written about this, called Strum-Liouville Theory in the most general case about eigenvalues and eigenfunctions. But that is not important for us, we can answer that question posted above with a little work.
Now there are three cases .
We will consider each case respectively.
Case 1: Then the general solution to is
We need to also satisfy the endpoints, thus,
The above is a linear system for it turns out that the only solution is because the determinant is non-zero (that is Cramer's Rule).
Which means that is the only solution to . Ah! But that is a trivial solution, hence there are no eigenfunctions for .
Case 2: Then the general solution is . We need to satisfy the conditions:
The first equation implies but then by the second equation. Again a trivial solution. Thus is not an eigenvalue of .
Case 3: The solution is given by . This leads to the linear systems:
The first equation forces which by the second equation means:
But does that mean ?
No! Because if then can be anything.
So when there are non-trivial solutions! In fact, those solutions are for any .
In conclusion, we have showed in detail that the eigenvalues to equation are:
And the eigenfunctions are (up to a multiplicative constant):
Just note, there are infinitely many eigenvalues for that equation. But soon we will see why this observation is extremely extremely important to the elegant solution of the heat equation.