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I'm trying to solve an ODE with DSolve. The problem I'm facing is that in the ODE there appears an implicit function of the variable r*. The equation I'm trying to solve is

-q''[r*] + (-2 (a/r^3) (1 - a/r)^3 - w^2) q[r*] == 0

with a and w parameters and r*[r]=2 a Log[r-a]-a^2/(r-a)+r (tortoise coördinate), which cannot be inverted such that I cannot write my ODE only in terms of r*. Is it possible to solve the equation by somehow using the implicit function relation?

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    $\begingroup$ r* means r times (whatever follows) in Mathematica. It seems you mean it to be either a function like rstar[r] or a variable rstar. Ultimately, do you want q as a function of r or of rstar? The latter should be theoretically possible if you write r[rstar] instead of r and use the implicit equation rstar==2 a Log[r[rstar]-a]-a^2/(r[rstar]-a)+r[rstar]. Probably DSolve can't solve it, I'd guess. But you could use NDSolve or ParametricNDSolve on numeric values for a and w. $\endgroup$
    – Michael E2
    May 21 at 13:54
  • $\begingroup$ You can use Format[rstar] = Superscript["r", "*"]; to change the display of rstar in output. $\endgroup$
    – Bob Hanlon
    May 21 at 16:46

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