It depends, to solve some types of differential equations you need an obvious solution. Can you show the differential equation you need to solve?
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.