Let be a real constant. consider the equation with boundary conditions and . for certain discrete values of , this equation can have non-zero solutions. find the three smallest values of for which this is the case.

This linear ODE has characteristic equation



with roots , which gives solutions of the form



There are three cases to consider:

(1) If , then the solution will be exactly what we see above. However, the initial conditions force both .

(2) If , we're left with



where is added to the solution set to account for a second solution that is linearly independent of the first solution. Again, we get .

(3) If , then the square root introduces a factor of that admits the solution



In this case, we arrive at , and from the second condition we get



In order that , we require that , where is any integer. Solving for , we get



When , we arrive at , but remember that we're assuming that , so logically the three smallest values of that are allowed occur for . (, so we can just look at positive integers .)

Unfortunately, I'm not sure exactly what's going on next. Checking with a computer, the solution is supposed to be



(Again, not sure why this is the case, but let's move on.) When , we have the least values, which are, respectively,