For each pair of real values (a,b) you'll get a unique solution corresponding to the initial conditions y(0) = a, y'(0) = b. If you let y1(x) be the solution corresponding y1(0) = 1, y1'(0) = 0, and y2(x) the solution corresponding to y2(0) = 0, y2'(0) = 1, then you can show that general solution for (a,b) is y(x) = a*y1(x) + b*y2(x).

You can also see this as part of the larger subject of Linear Algebra, which introduces the concept of "dimension" more generally and proves theorems that under certain conditions the dimensions of something (that something being a vector space) the dimension is well defined and how to calculate it. Here the dimension of the solution space is 2.