# Singular perturbation problem

Find all solutions of the equation u x^5 + x + 1 = 0 in an perturbative expansion of the small parameter u.

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#### Solution Preview

The equation:

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 ....

therefore:

x = -1 + u - u^2 ...

#### Solution Summary

A detailed solution containing an explanation of the singular perturbation method is given.

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