Answer the following problems. Show all your working.
Consider the equation
εx^{3} + x^{2} - x - 6 = 0, ε > 0. (1)
1. Apply a naive regular perturbation of the form
x ^{∞}∑_{n=0} x_{n}ε^{n} 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.