How to Invert a Monotonic Function that Contains an Integral?

Question

Consider $y=b(c)$ where the function $b$ is continuous, strictly increasing in its argument. So that there exists an inverse. However, $b(c)$ is quite complicated and has an integral: $$ b(c) = \int_{c}^{1} g(x) dx ~ . $$

The problem is to obtain the inverse of $b$:

$$ c = b^{-1}(y) $$

Answer 1

One thing you can say is that $b^{-1}(y) = c$ is a solution of the initial value ODE problem $$ \eqalign{g(c)\; c' &= -1\cr c(0) &= 1 \cr}$$ As mentioned in the comments, it's unlikely to have closed-form explicit solutions. But, given good numerical ODE solvers, this may be a good way to get numerical results.

EDIT: As requested, here's an example. I'll use Maple, and take $g(x) = \dfrac{2 + \sin(x)}{1+x^2}$.

g:= x -> (2+sin(x))/(1+x^2):
S:= dsolve({g(c(y)) * diff(c(y),y) = -1, c(0) = 1},c(y), numeric):
S(1);

$$[y = 1., c(y) = .428444801994959]$$

plots[odeplot](S, [y,c(y)],y=-1.5..4);