Example 4.4 We substitute a perturbation series I( T, () = IO(7) + EXI (T) + EE2 (T) + ... into equations (4.1), (4.2) and get d' co 1 +E- ad 12 d72 +... = (1 + cro + ( x] + ... )?' Io(0) + er (0) +er, (0) +.. = dro dT (O)+E- dT dT (0) +... = 1. In order to equate coefficients of like powers of e in the differential equation, we must first change the right-hand side into a series in powers of c. We can do this quickly by recalling the binomial series: 2! which converges for (2|