I'm solving the following PDE:

DSolve[F''[ξ] + (Ω^2 - 
  M^2 Exp[2 α ξ]) F[ξ] == 0, F[ξ], ξ]

And Mathematica gives a general solution in terms of modified Bessel functions:

$c_1\cdot\frac{(-1)^{-\frac{i\Omega}{2\alpha}}}{\Gamma(1-\frac{i\Omega}{\alpha})}I_{-\frac{i\Omega}{\alpha}}(\frac{M}{\alpha}e^{\alpha\xi})+ c_2\cdot\frac{(-1)^{\frac{i\Omega}{2\alpha}}}{\Gamma(1+\frac{i\Omega}{\alpha})}I_{\frac{i\Omega}{\alpha}}(\frac{M}{\alpha}e^{\alpha\xi})$

Rather than just:

$c_1\cdot I_{-\frac{i\Omega}{\alpha}}(\frac{M}{\alpha}e^{\alpha\xi})+ c_2\cdot I_{\frac{i\Omega}{\alpha}}(\frac{M}{\alpha}e^{\alpha\xi})$

And the latter is also a valid solution of the PDE. Any idea why these extra factors aren't absorbed into the $c$'s?

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Presumably, because it doesn't have a rule for that simplification. If you want it simplified, use a rule to do that.

DSolve[F''[\[Xi]] + (\[CapitalOmega]^2 - 
        M^2 Exp[2 \[Alpha] \[Xi]]) F[\[Xi]] == 0, F[\[Xi]], \[Xi]][[1]] /. 
 C[n_]*z_?(FreeQ[#, \[Xi]] &) :> C[n]

enter image description here


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