# Implicit function in an ODE when using DSolve

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?

• 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. May 21 at 13:54
• You can use Format[rstar] = Superscript["r", "*"]; to change the display of rstar in output. May 21 at 16:46