Problems on Power Series Solutions of DE's

Find a power series solution for each of the following differential equations by first finding a recurrence relation for c_n. Then use the series on page 628 of your textbook to identify the series solution in terms of familiar elementary functions. (You may and should use Maple or some other method to check your work.)

1. (H) y' = 2y - x
2. (H) (4x + 1)y' = - 4y
3. (H) y'' + 9y = x

4. The differential equation (x - 2)y'' = y cannot be solved in terms of elementary functions, but it can be solved in the form of a series.

   (a)

   Find a power series solution for (x - 2)y'' = y by first finding a recurrence relation for c_n. Write the first ten terms of the series.

   (b)

   Suppose we have the initial conditions y(0) = 1, y'(0) = 1. Graph the fourth degree polynomial approximation to your solution over the interval [0, 1]. Use your graph to estimate where the solution reaches its maximum on the interval [0, 1].

   (c)

   If you use a sixth-degree polynomial approximation of your solution, will your answer change? (Try it.) What about tenth-degree?

   (d)

   Are your approximations better near the left side of the interval or the right? Why?

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Problems on Power Series Solutions of DE's

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