Ah yes that was a slip up. You're right, it should read the "space of all differentiable functions," or maybe "all complex differentiable functions" for extra precision.
The space of all differentiable linear functions is a vector space, and the derivative is a linear operator on that space (so functions and derivatives follow most of the same rules as vectors and matrices). If D is the derivative operator and x is a function of t, the equation x''=x can be written as D^2 * x = x, or 0 = (D^2 - I)x = (D + I)(D - I)x. The dimension of the kernel of (D - I) is 1, and so is the dimension of the kernel of (D + I). So the space of solutions to the original problem has dimension 0, 1, or 2. But you can find two solutions, namely cos(i * t) and sin(i * t). Those solutions are linearly independent, so they span 2 dimensions. So all solutions are going to be of the form A * cos(i * t) + B * sin(i * t). The case for x''=ax is similar.