« Section 8.6

Section 8.7

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1. (a) \displaystyle{x = a_0\sum_{n=0}^\infty \frac{3^nt^n}{n!} = a_0e^{3t}}

(c) \displaystyle{x = a_0 + (a_0 - 1)\sum_{n=1}^\infty \frac{t^n}{n!} = 1 + (a_0 - 1)e^t}

2. (a) For t in (-\infty, \infty) ,

\displaystyle{x = a_0\left(1 - \frac{1}{6}t^3 + \frac{1}{180}t^6 - \cdots\right) + a_1\left(t - \frac{1}{12}t^4 + \frac{1}{504}t^7 - \cdots\right).}

(c) For t in (-\infty, \infty) ,

\displaystyle{x = a_0\left(1 - \frac{1}{2}t^2 + \frac{1}{8}t^4 - \cdots\right) + a_1\left(t - \frac{1}{3}t^3 + \frac{1}{15}t^5 - \cdots\right).}

(e) For t in (-1, 1) ,

\displaystyle{x = a_0\left(1 + \frac{1}{2}t^2 + \frac{7}{24}t^4 + \cdots\right) + a_1\left(t + \frac{1}{2}t^3 + \frac{13}{40}t^5 + \cdots\right).}

5.

\mathbf{r} Polynomial solution
0 x_1(t) = 1
1 x_2(t) = t
2 x_1(t) = 1 - 2t^2
3 x_2(t) = t - \frac{2}{3}t^3
4 x_1(t) = 1 - 4t^2 + \frac{4}{3}t^4
5 x_2(t) = t - \frac{4}{3}t^3 + \frac{4}{15}t^5

7. (c)

Legendre polynomials
P_0(t) = 1
P_1(t) = t
P_2(t) = -\frac{1}{2} + \frac{3}{2}t^2
P_3(t) = -\frac{3}{2}t + \frac{5}{2}t^3
P_4(t) = \frac{3}{8} - \frac{15}{4}t^2 + \frac{35}{8}t^4
P_5(t) = \frac{15}{8}t - \frac{35}{4}t^3 +\frac{63}{8}t^5

This entry was posted on 22 August 22 2007 at 9:01 am and is filed under Section 8.7. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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