Is there syntax that causes Mathematica solve integral equations like the one below (I know that I can transform it to a differential equation. However, I would like to know if Mathematica solves equations in integral form.)?
$$ \int_{r_0}^{x} f(z) dz = \frac{kf(x)-L}{a} $$
From the documentation of DSolve (emphasis mine):
DSolvecan solve ordinary differential equations (ODEs), partial differential equations (PDEs), differential algebraic equations (DAEs), delay differential equations (DDEs), integral equations, integro-differential equations, and hybrid differential equations.
So you can just pass the equation to DSolve, as you would with a differential equation:
DSolve[Integrate[f[z], {z, r0, x}] == (k f[x] - L)/a, f[x], x]
(* {{f[x] -> (E^((a (-r0 + x))/k) L)/k}} *)
