You're not trying to find a general solution, as you were already given it. You're looking for a way to solve for and

First, you take the derivatives:

One of the first things you can conclude is that cannot possibly be zero, or else it will render your general solution to be zero, which is obviously not true, this makes the resulting equation:

Now we group all of the terms together:

Now, we know that cosine and sine are linearly independent, which means that the only way that a linear combination of these two functions can add up to zero is when the coefficients of each one are both equal to zero, and so:

Now, before we go any further, we must find alpha, which is solved in the next coefficient:

Now we see alpha squared:

Plug and simplify:

Which is:

So, omega equals:

Somebody else will have to do phi, I have to go to class.