  We present a self-contained formalism for analyzing scale invariant
differential equations. We first cast the scale invariant model into its
equidimensional and autonomous forms, find its fixed points, and then obtain
power-law background solutions. After linearizing about these fixed points, we
find a second linearized solution, which provides a distinct collection of
power laws characterizing the deviations from the fixed point. We prove that
generically there will be a region surrounding the fixed point in which the
complete general solution can be represented as a generalized Frobenius-like
power series with exponents that are integer multiples of the exponents arising
in the linearized problem. This Frobenius-like series can be viewed as a
variant of Liapunov's expansion theorem. As specific examples we apply these
ideas to Newtonian and relativistic isothermal stars and demonstrate (both
numerically and analytically) that the solution exhibits oscillatory power-law
behaviour as the star approaches the point of collapse. These series solutions
extend classical results. (Lane, Emden, and Chandrasekhar in the Newtonian
case; Harrison, Thorne, Wakano, and Wheeler in the relativistic case.) We also
indicate how to extend these ideas to situations where fixed points may not
exist -- either due to ``monotone'' flow or due to the presence of limit
cycles. Monotone flow generically leads to logarithmic deviations from scaling,
while limit cycles generally lead to discrete self-similar solutions.
