Construct an example of a second-order, non-constant coefficient, linear, homogeneous ODE. Then answer the following questions. (a) When is a unique solution guaranteed? (b) Solve this equation using power series. (c) Solve this equation again using a different method. Must your solutions match? Why or why not? If so, show that your solutions are equivalent. (d) Let f be a continuous function. Suppose the ODE you constructed is the associated homogeneous equation of some non-homogeneous ODE of the from G(x, y, y', y") = f(x) (i.e. just add f(x) as the non-homogeneous part). Is it possible to use any of our non- homogeneous technqiues to solve this non-homogeneous ODE? If so, solve it. If not, justify why each method we've covered wouldn't apply.

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