At any age we, being somewhat children, ask that. Here is the answer to such a question related to Maple. Let us consider the ODE

ode:=(2x^2-x-1)*y''(x)-(4x^3+x-2)*y'(x)+(4x^3+2x^2-2x+2)*y(x)=0;

and its general solution done with Maple

dsolve(ode);

y(x) = _C1*exp(x^2)+_C2*exp(x)*x.

The first thought is that Maple, mimicking the solution done by hand, finds a particular solution of the equation in the form y*(x)=exp(a*x^2+b*x). It fits the values of a and b to satisfy the ODE under consideration: y*(x)=exp(x^2). After that Maple finds its general solution in the form C(x)*y*(x),               using the method of variation of constant. It remains to wonder how all that things are programmed in Maple. 

The second thought is to execute the dsolve(ode) command at the infolevel[dsolve]:=5 in order to look into Maple work. The output is as follows.

Methods for second order ODEs:

--- Trying classification methods

---trying a quadraturechecking

if the LODE has constant coefficientschecking

if the LODE is of Euler type

trying a symmetry of the form [xi=0, eta=F(x)] 

 testing BRANCH 1 -> 

 testing BRANCH 2 -> 

 testing BRANCH 3 -> 

 testing BRANCH 4 -> 

 testing BRANCH 5 -> 

 testing BRANCH 6 ->

checking if the LODE is missing 'y'

 ->Trying a Liouvillian solution using Kovacic's algorithm 

 A Liouvillian solution exists 

 Reducible group (found an exponential solution) 

 Reducible group (found another exponential solution)

<- Kovacic's algorithm successful

y(x) = _C1*exp(x^2)+_C2*exp(x)*x

The googling of "Kovacic" brings a footballer Mateo Kovacic and the artilcle

http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/jjk_algorithm_scan.pdf

It appears that Maple uses differential algebra (for example, see http://en.wikipedia.org/wiki/Differential_algebra ) to this end.