# Perturbation Theory

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Find the real root of the equation x^5 + ex - 1= 0 approximately to O(^2) using perturbation theory.

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#### Solution Preview

Solution. We assume that the root x is of the following form

x=a+e*b

So, x^5=(a+e*b)^5=a^5+5*e*a^4*b+10*e^2*a^3*b^2+10*e^3*a^2*b^3+

5*e^4*a*b^4+b^5

So, the equation x^5+ex-1=0 is becoming

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#### Solution Summary

Real roots are found using perturbation theory. The real root of the equations are analyzed.

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