Consider the equation

ex³ + x² - x - 6 = 0, e > 0 (1)

1. Apply a naive regular perturbation of the form

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 e ! 0+. Again, derive a three-term approximation this third case.

3. Solve (1) numerically for e = 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.