I have a problem with the DSolve operator. It just gives me the initial ODE as an answer, however, obviously, I need to find the answer h[s] as a function of f[s] and its powers (or whatever else).

r[s_, t_] := {s, f[s]*Cos[t], h[s]*Sin[t]}
r1[s_, t_] := D[r[s, t], s] // Simplify
r2[s_, t_] := D[r[s, t], t] // Simplify
g11[s, t] := r1[s, t].r1[s, t] // Simplify
g22[s, t] := r2[s, t].r2[s, t] // Simplify
g12[s, t] := r1[s, t].r2[s, t] // Simplify

detG := g11[s, t] g22[s, t] - g12[s, t] g12[s, t] // Simplify
derivG = 1/2*D[detG, s] // FullSimplify

DSolve[derivG == 0, h[s], s]

So, how does one force Mathematica to solve the eqn? Could someone point me on some mistakes if there are any? Thanks in advance.

I have a problem with the DSolve operator. It just gives me the initial ODE as an answer, however, obviously, I need to find the answer h[s] as a function of f[s] and its powers (or whatever else). It is an differential-algebraic equation, but I do not have any connection between f[s] and h[s], they're independent.

r[s_, t_] := {s, f[s]*Cos[t], h[s]*Sin[t]}
r1[s_, t_] := D[r[s, t], s] // Simplify
r2[s_, t_] := D[r[s, t], t] // Simplify
g11[s, t] := r1[s, t].r1[s, t] // Simplify
g22[s, t] := r2[s, t].r2[s, t] // Simplify
g12[s, t] := r1[s, t].r2[s, t] // Simplify

detG := g11[s, t] g22[s, t] - g12[s, t] g12[s, t] // Simplify
derivG = 1/2*D[detG, s] // FullSimplify

DSolve[derivG == 0, h[s], s]

So, how does one force Mathematica to solve the eqn? Could someone point me on some mistakes if there are any? Thanks in advance.