When told to show that something is a solution of a differential equation, then asked to determine the general solution, how do I do it?
Unless you are given other information, there is no way to go from a given solution to the general solution.
Now, if the differential equation is linear second order equation you can use a given solution to reduce to a first order equation and solve that for a second, independent, solution and so write the general solution as a linear combination of the two solutions.