So "second order linear ordinary differential equation with rational coefficients" means that f should satisfy an equation of this type: A(z)f''(z)+B(z)f'(z)+C(z)f(z)=D(z) where the functions A(z), B(z), C(z), and D(z) are rational FUNCTIONS. If all of the four are supposed to be rational NUMBERS, then that's a constant coeff equation. If you substitute exp(rz) in the associated homogeneous equation, you will get a quadratic equation for r (the "characteristic equation" is what it is called, I think). Then the solutions will all be linear combinations of these exponentials (with something simple happening if the root is multiple). But I am fairly sure the dilogarithm is NOT a linear combination of exponentials (hey: the radius of convergence is finite!).
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