Consider the second-order differential equation
Find all values for the constant such that the equation has a nontrivial solution such that
For each such , determine the corresponding solution .
If , then
The condition then implies , so only the trivial solution is possible.
If , then then is of the form
Therefore, and since . Hence the solutions are of the form
Then, implies , and implies
where is an integer, or . But, if then we have the trivial solution.
If , then is of the form
Therefore, and since . Hence, the solutions are of the form
The condition implies and the condition implies
But this latter solution is only possible if , contradicting that . Hence, only the trivial solution is possible for .
Putting this all together, the only nontrivial solution is for , and is given by