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de2de:chapter-1 [2012/06/25 11:10] (current)
dcs created
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 +====Differential Equations to Difference Equations====
  
 +===Answers for selected problems===
 +
 +**Chapter 1**
 +
 +----
 +
 +**Section 1.1**
 +
 + ​4. ​    ​(a) ​ $ y = 4(x - 1) + 2  $ 
 +
 + ​(b) ​ $ y = 2(x - 1) + 2  $ 
 +
 + ​%%(c)%% ​  $ y = 3(x - 1) + 1  $ 
 +
 + ​(d) ​ $ y = 4(x - 2) + 4  $ 
 +
 + ​5. ​ For  $ 6  $ digits, ​ $ n = 5624  $. 
 +
 + ​6. ​   $ n \ge 201  $ 
 +
 + ​7. ​ (a)  $ p = 101  $ 
 +
 + ​(b) ​ $ |m_n - 2| < 0.01  $ when  $ n > p  $  ​
 +
 + ​8. ​ (a)  $ a_{20} = 0.9996 ​ $ to four decimal places. ​
 +
 + ​%%(c)%% ​ $ a_{20} = 2.6533 ​ $ to four decimal places. ​
 +
 + ​(d) ​ $ a_{20} = 41.1032 ​ $ to four decimal places.
 +
 +----
 +
 +**Section 1.2**
 +
 +1.    (a) $ \displaystyle{a_n = \frac{1}{3^n},​ n=0,​1,​2,​\ldots}$
 +
 +(b) $ \displaystyle{a_n = \frac{1}{n},​ n=1,2,3, \ldots}$
 +
 +%%(c)%% $ \displaystyle{a_n = \frac{2n+1}{n+1},​ n=0,​1,​2,​\ldots}$
 +
 +(d) $ \displaystyle{a_n = \frac{(-1)^n}{2n+1},​ n=1,​2,​3,​\ldots}$
 +
 +2. (a) Converges: $ \displaystyle{\lim_{n \to \infty}\frac{1}{3^n} = 0}$
 +
 +(b) Diverges: $ \displaystyle{\lim_{n \to \infty}\pi^n = \infty}$
 +
 +%%(c)%% Converges: $ \displaystyle{\lim_{n \to \infty}\frac{3n-1}{2n+6} = \frac{3}{2}}$
 +
 +(d) Diverges
 +
 +(e) Diverges: $ \displaystyle{\lim_{n \to \infty}\frac{3n^4-6n^3+1}{5n^3+n^2+2}=\infty}$
 +
 +(f) Converges: $ \displaystyle{\lim_{n \to \infty}\frac{2n^5-3n^2+23}{7n^5+13n^4-12} =  \frac{2}{7}}$
 +
 +(g) Converges: $ \displaystyle{\lim_{n \to \infty}\frac{45-16n^2}{13+5n+6n^3} = 0}$
 +
 +(h) Converges: $ \displaystyle{\lim_{n \to \infty}\frac{3n+1}{\sqrt{4n^2+1}}=\frac{3}{2}}$
 +
 +(i) Diverges: $ \displaystyle{\lim_{n \to \infty}(-2)^{2n+1} = -\infty}$
 +
 +(j) Divergs: $ \displaystyle{\lim_{n \to \infty}\frac{10-16n^3}{1+n^2}=-\infty}$
 +
 +(k) Converges: $ \displaystyle{\sqrt{\frac{3n^2+n-6}{5n^2+16}}=\sqrt{\frac{3}{5}}}$
 +
 +(l) Converges: $ \displaystyle{\lim_{n \to \infty}\frac{(-1)^n}{5^n}=0}$
 +
 +3. $ \displaystyle{\lim_{n \to \infty}\frac{\sin(n)}{n}=0}$
 +
 +4.  (a) $ a_1 = 2 $, $ a_2 = 2.25 $, $ a_3 = 2.3704 $, $ a_4 = 2.4414 $, and $ a_5 = 2.4883 $, where the last three have been rounded to four decimal places.
 +
 +(e) $ a_{74} > 2.7 $, but $ a_n < 2.7 $ for $ n < 74 $.
 +
 +(f) $ n = 135 $
 +
 +5. (e) $ n = 13 $
 +
 +(f) $ n = 41 $
 +
 +7. %%(c)%% $ \displaystyle{a_9 = a_{10} = \frac{1562500}{567}} $
 +
 +11. (a) $ \displaystyle{N = \left\lfloor \frac{1}{\epsilon}\right\rfloor} $; for $ \epsilon = 0.001 $, $ N = 1000 $
 +
 +(b) $ \displaystyle{N = \left\lfloor \frac{\log(\epsilon)}{\log(0.98)}\right\rfloor} $; for $ \epsilon = 0.001 $, $ N = 341 $
 +
 +%%(c)%% $ \displaystyle{N = \left\lfloor \frac{1}{\sqrt{\epsilon}}\right\rfloor} $; for $ \epsilon = 0.001 $, $ N = 31 $
 +
 +(d) $ \displaystyle{N = \left\lfloor \frac{1}{\sqrt[3]{\epsilon}}\right\rfloor} $; for $ \epsilon = 0.001 $, $ N = 10 $
 +
 +----
 +
 +**Section 1.3**
 +
 +1. (a) $ \displaystyle{\sum_{n=1}^\infty \left(\frac{1}{3}\right)^{n-1} = \frac{3}{2}}$
 +
 +%%(c)%% $ \displaystyle{\sum_{n=1}^\infty \left(\frac{2}{5^n}\right) = \frac{1}{2}}$
 +
 +(e) $ \displaystyle{\sum_{n=1}^\infty 7\left(\frac{1}{3}\right)^n\frac{2}{5}^{n-1} = \frac{35}{13}}$
 +
 +(g) Does not converge
 +
 +(i) Does not converge
 +
 +(k) Does not converge
 +
 +(m) $ \displaystyle{\sum_{n=1}^\infty \sin(n\pi) = 0}$
 +
 +(n) Does not converge
 +
 +3. (a) $ \pi $
 +
 +(b) $ 99 $
 +
 +5. (a) $ 10 $ billion dollars
 +
 +%%(c)%% $\displaystyle{\frac{A}{1 - r}}$ dollars
 +
 +7. $70$ meters; $\displaystyle{frac{50}{3}}$ meters ​
 +
 +9. $\displaystyle{\frac{4}{9}}$
 +
 +11. $ 83 $ terms for a partial sum larger than $ 5 $; $ 12,367 $ terms for a partial sum larger than $ 10 $
 +
 +----
 +
 +**Section 1.4**
 +
 +1. (a) $ x_0 = 10 $, $ x_1 = 14 $, $ x_2 = 18 $, $ x_3 = 22 $, $ x_4 = 26 $, $ x_5 = 30 $ 
 +
 +%%(c)%% $ x_0 = 40 $, $ x_1 = 60 $, $ x_2 = 100 $, $ x_3 = 180 $, $ x_4 = 340 $, $ x_5 = 660 $ 
 +
 +(e) $ x_0 = 2 $, $ x_1 = 3 $, $ x_2 = 5 $, $ x_3 = 8 $, $ x_4 = 13 $, $ x_5 = 21 $ $ x_6 = 34 $ 
 +
 +2. (a) $ x_n = 5 \cdot 2^n $, $ n = 0, 1, 2, \ldots $; $ x_{10} = 5120 $ 
 +
 +%%(c)%% $ x_n = 32.5(1.8)^n - 12.5 $, $ n = 0, 1, 2, \ldots$; $ x_{10} = 11,591.5 $, rounded to one decimal place 
 +
 +(e) $ \displaystyle{x_n = \frac{10}{3} \cdot 4^n - \frac{4}{3}}$,​ $ n = 0, 1, 2, \ldots $; $ x_{10} = 3,495,252 $
 +
 +3. (a) $ w_{n+1} = 1.03w_n $, $ n = 0, 1, 2, \ldots $
 +
 +(b) $ w_n = 350(1.03)^n $, $ n = 0, 1, 2, \ldots $
 +There will be $ 545 $ weasels in $ 15 $ years.
 +
 +(d) $ 24 $ years 
 +
 +(e) $ \displaystyle{\lim_{n \to \infty}w_n = \infty} $
 +This says the population will grow without any upper bound to its size, which is not possible due to the physical limitations of the habitat.
 +
 +5. (a) $ w_n = 1.03w_n - 6 $, $ n = 0, 1, 2, \ldots $
 +
 +(b) $ w_n = 150(1.03)^n + 200 $, $ n = 0, 1, 2, \ldots $
 +There will be $ 434 $ weasels in $ 15 $ years.
 +
 +%%(c)%% $ \displaystyle{\lim_{n \to \infty}w_n = \infty}$
 +This says the population will grow without any upper bound to its size.
 +
 +(d) The population will double in $ 41 $ years.
 +
 +7. (a) $ 155.30 $ grams, rounded to two decimal places
 +
 +(b) About $ 686 $ years
 +
 +%%(c)%% About $ 2,279 $ years
 +
 +9. (a) $ \displaystyle{T_{n+1} = \frac{17}{19}T_n + \frac{140}{19}}$,​ $ n = 0, 1, 2, \ldots $
 +
 +(b) $ \displaystyle{T_n = 95\left(\frac{17}{19}\right)^n + 70}$, $ n = 0, 1, 2, \ldots $
 +
 +%%(c)%% $ 75.9^\circ$ F, rounded to one decimal place
 +
 +(d) $ \displaystyle{\lim_{n \to \infty}T_n = 70^\circ}$ F
 +
 +(f) No
 +
 +11. $ 103.8^\circ $ F, rounded to one decimla place
 +
 +15. $ x_n = 4n + 10 $, $ n = 0, 1, 2, \ldots, 10 $ 
 +
 +----
 +
 +**Section 1.5**
 +
 +1. (a) $ w_{n+1} = w_n + 0.00003w_n(1000 - w_n) $, $ n = 0, 1, 2, \ldots $ 
 +
 +%%(c)%% The population will double in $ 42 $ years and triple in $ 102 $ years.
 +
 +(d) From the plot, it appears that $ \displaystyle{\lim_{n \to \infty}w_n = 1000}$ ​
 +
 +3. (a) $ p_{n+1} = 1.045p_n $, $ n = 0, 1, 2,  \ldots $
 +
 +%%(c)%% The population will double in $ 16 $ years and triple in $ 25 $ years.
 +
 +(d) $ \displaystyle{\lim_{n \to \infty}p_n = \infty}$
 +
 +(e) $ 52 $ years
 +
 +4. (a) It appears that $ \displaystyle{\lim_{n \to \infty}r_n = 10,000}$
 +
 +%%(c)%% There appears to be a limiting cycle of period $ 2 $, with values alternating between $ 6,403 $ and $ 11,930 $.
 +
 +(e) There appears to be a limiting cycle of period $ 8 $, with values cycling thorough $ 4851 $, $ 11245 $, $ 7661 $, $ 12248 $, $ 5198 $, $ 11588 $, $ 6876 $, and $ 12375 $.
 +
 +7. (a) $ \displaystyle{\lim_{x \to \infty}x_n = \infty}$
 +
 +%%(c)%% It appears that $ \displaystyle{\lim_{n \to \infty}x_n = 0.7391}$ to four decimal places.
 +
 +(e) It appears that $ \displaystyle{\lim_{n \to \infty}x_n =\sqrt{2}}$.
 +
 +(g) It appears that $ \displaystyle{\lim_{n \to \infty}x_n = \infty}$.
 +
 +(i) There appears to be a limiting cycle of order $ 2 $.
 +
 +(k) The limiting behavior, if any, is not clear.
 +
 +----