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