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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?

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    $\begingroup$ Isn't your $A$ complex? What about the real and imaginary parts of $e^{ik^3/3}$?. There you have two real independent solutions. $\endgroup$ – GReyes Feb 11 at 8:20
  • $\begingroup$ Are we working over reals or complexes? $\endgroup$ – mathreadler Feb 11 at 8:21
  • $\begingroup$ @GReyes, it's complex, yes. Are you sure it gives two independent solutions? $\endgroup$ – Yuriy S Feb 11 at 8:21
  • $\begingroup$ @mathreadler, the variable is real $\endgroup$ – Yuriy S Feb 11 at 8:22
  • $\begingroup$ Same as needing a linear combination with sin and cos solving real case but just one complex exponential in complex case (if you remove "times x"). $\endgroup$ – mathreadler Feb 11 at 8:22

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