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The eigenfunction $u$ of a regular Sturm–Liouville problem is either a real valued function or a complex constant multiplied by a real valued function.

Given hint: Observe that if $u$ is an eigenfunction, then so is $\bar{u}$ corresponding to the same eigenvalue. Then use the fact that eigenvalues are simple to show that $u=c\bar{u}$ where $c$ is a constant.

Notes: The regular Sturm–Liouville problem is defined here.

I have also proved previously that for the regular Sturm–Liouville problem the resulting eigenvalues are real and the eigenfunctions form a basis. Please tell me if these properties are related to the problem, and how can I use them for the basis of a proof.

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You do not need to know that the eigenfunctions form a basis in order to prove what you want. The other properties are relevant, though. As the hint stated, if $u,v$ are non-trivial eigenfunctions with the same eigenvalue of a regular problem, then there is a non-zero constant $C$ such that $u=Cv$. The coefficients of a regular problem are also real, the eigenvalues are real, and the endpoint conditions are real. So, if you conjugate the eigenvalue equation (including endpoint conditions) for $u$, then you end up with an identical equation for $\overline{u}$. So you can conclude that there is a non-zero constant $C$ such that $\overline{u}=Cu$.

Assuming $u$ is a non-trivial solution, the constant $C$ must satisfy $|C|=1$ because $|u|=|\overline{u}|=|C||u|$ implies $|C|=1$. And,$$ u+\overline{u}=(1+C)u, \;\; i(u-\overline{u})=i(1-C)u $$ are real eigenfunction solutions. One of these is non-trivial because $u$ is non-trival. So there is a non-zero complex constant $D$ such that $Du$ is real.