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Answer the following problems. Show all your working.

Consider the equation

εx3 + x2 - x - 6 = 0, ε > 0.                  (1)

1. Apply a naive regular perturbation of the form

x n=0 xnεn as ε→0+

do derive a three-term approximation to the solutions of (1).

2. The above perturbation expansion should only give you an approximation for 2 of the roots. Apply a leading order balance argument to device suitable expansions for the other root, again in the limit ε→0+ . Again, derive a three-term approximation this third case.

3. Solve (1) numerically for ε = 0.01 (use Matlab or Maple or something). Use your three-term approximation for the three roots found in Q1 and 2 and provide the error (in terms of a percentage) in each case.

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