I will say that paturbation method should not be seen as a purely approximate stand alone analytical method for all problems. Most authors explains it as either a purely analytical method, or a combination of numerical and analytical techniques depending on the complexity of the resulting order of resulting equations. See the attached extract from Hinch's book on the subject.

While using these approximate methods surely you will get the linear systems if possible you can get the exact solutions of these system otherwise you can compute the solution with the help of any approximate method. Usually the purpose of these methods are to linearize the problems and then find the solutions exactly like, eigen function, separation of variable, Laplace transform, Fourier, Greens function or many other where applicable. Symmetries are also useful if workable.