Case 1: correct.
Case 2: you have come to the right result but via the wrong method. For case 2 you have:
so integrating twice gives
and to satisfy BCs hence no solution as you concluded (for the wrong reasons).
Case 3: you have correctly found that:
so then, after finding the auxiliary equation, we find the solution for is
where so the general solution is given by:
Applying the initial conditions on u(x,0) and u'(x,0) we have
In the first of these summations you can find by multiplying by on both sides and then integrating wrt x from 0 to 3. Doing this should give you that:
I may have made arithmetic errors here so double check the result in case.
For the second summation it's exactly the same procedure to give you . At the end substitute back into your general solution the results for and and you're done!
An alternative, and easier way would have been to transform the problem to Fourier space and then solve.
Hope that helped!