For linear ODEs the number of linearly independent solutions is equal to the order of the equation. However, it's not clear to me how to recover the same number of solutions in I'm using Fourier transform to solve the equation.
For example, the Airy equation (with unspecified boundary conditions):
$$y''(x)-x y(x)=0$$
Using Fourier transform, we get:
$$-k^2 \hat{y}(k)-i \hat{y}'(k)=0$$
Which is a separable first order ODE.
So we get:
$$\hat{y}(k)=A e^{i k^3/3}$$
Where we only have one integration constant. Inverse transform doesn't lead to any additional constants.
So, what should I do to recover two linearly independent solutions of the original equation?
Should the boundary conditions always be specified before applying Fourier transform?