Find all solutions of the equation u x^5 + x + 1 = 0 in an perturbative expansion of the small parameter u.© BrainMass Inc. brainmass.com March 5, 2021, 1:52 am ad1c9bdddf
u x^5 + x + 1 = 0 (1)
is of fifth degree and therefore has five solutions in the complex plane. If we write down a perturbation series for the solutions of the form:
x = x0 + x1 u + x2 u^2 + x3 u^3 + ....
and substitute this in (1) and equate equal powers of u, what we get is a perturbation series for a single solution:
u (x0 + x1 u + x2 u^2 + .... )^5 + x0 + x1 u + x2 u^2 + ... + 1 = 0 ------>
x0 + 1 + (x1 + x0^5) u + (x2 + 5 x0^4 x1) u^2 +.... = 0 ------->
x0 = -1; x1 = 1; x2 = -1 ....
x = -1 + u - u^2 ...
A detailed solution containing an explanation of the singular perturbation method is given.