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 — de2de:chapter-7 [2012/06/25 11:35] (current)dcs created 2012/06/25 11:35 dcs created 2012/06/25 11:35 dcs created Line 1: Line 1: + ====Differential Equations to Difference Equations==== + ===Answers for selected problems=== + + **Chapter 7** + + ---- + + **Section 7.1** + + 1. (a) $w + z = 1 + 3i$ + + %%(c)%% $3w - 2z = 13 - 26i$ + + (e) $zw = 22 + 29i$ + + (f) $\displaystyle{\frac{z}{w} = -\frac{34}{25} + \frac{13}{25}i}$ ​ + + (g) $|z| = \sqrt{53}$ + + (i) $\Re(z - w) = -5$ + + (j) $\Im(3z + w) = 17$ + + + 2. (a) $\displaystyle{\Re\left(\frac{1}{i}\right) = 0}$, $\displaystyle{\Im\left(\frac{1}{i}\right) = -1}$ + + %%(c)%% $\displaystyle{\Re\left(\frac{3-4i}{-2+3i}\right) = -\frac{18}{13}}$,​ $\displaystyle{\Im\left(\frac{3-4i}{-2-3i}\right) = -\frac{1}{13}}$ ​ + + + 3. (a) $z = 3i$ + + %%(c)%% $\displaystyle{z = -\frac{1}{2\sqrt{2}} - \frac{1}{2\sqrt{2}}i}$ ​ + + + 4. (a) $|z| = 1$, $\mathrm{Arg}(z) = -\dfrac{\pi}{2}$ + + %%(c)%% $|z| = \sqrt{2}$, $\mathrm{Arg}(z) = \dfrac{\pi}{4}$ + + (e) $|z| = 4$, $\mathrm{Arg}(z) = \dfrac{\pi}{3}$ + + + 5. (a) $\left|w^2\right| = 9$, $\mathrm{Arg}\left(w^2\right) = \dfrac{\pi}{3}$, $\displaystyle{w^2 = \frac{9}{2} + \frac{9\sqrt{3}}{2}i}$ ​ + + %%(c)%% $|wz| = 6$, $\mathrm{Arg}(wz) = -\dfrac{\pi}{6}$, $\displaystyle{wz = 3\sqrt{3} - 3i}$ + + (e) $\left|\dfrac{z}{w^2}\right| = \dfrac{2}{9}$, $\mathrm{Arg}\left(\dfrac{z}{w^2}\right) = -\dfrac{2\pi}{3}$, $\displaystyle{\frac{z}{w^2} = -\frac{1}{9} - \frac{1}{3\sqrt{3}}i}$ ​ + + (f) $\left|w^5\right| = 243$, $\mathrm{Arg}\left(w^5\right) = \dfrac{5\pi}{6}$, $\displaystyle{w^5 = -\frac{243\sqrt{3}}{2} + \frac{243}{2}i}$ ​ + + + 6. If $z = \dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i$, then the roots are $1, z, z^2, z^3, z^4$, and $z^5$. + + + 8. %%(c)%% The square roots of $1 + \sqrt{3}i$ are $\sqrt{\dfrac{3}{2}} + \dfrac{1}{\sqrt{2}}i$ and $-\sqrt{\dfrac{3}{2}} - \dfrac{1}{\sqrt{2}}i$. + The square roots of $-9$ are $3i$ and $-3i$. + + ---- + + **Section 7.2** + + 1. (a) $\displaystyle{\lim_{n \to \infty}z_n = -1 + \frac{1}{2}i}$ ​ + + %%(c)%% $\displaystyle{\lim_{n \to \infty}z_n = 3}$ + + + 2. (a) $\displaystyle{\lim_{z \to i}(4z^3 - 6z + 3) = 3 - 10i}$ + + %%(c)%% $\displaystyle{\lim_{w \to 3i}\frac{w^2 + 9}{w - 3i} = 6i}$ + + + 3. (a) $f'(z) = 6z - 30z^4$ + + %%(c)%% $\displaystyle{f'​(z) = -2z(z - 4i)e^{-z^2} + e^{-z^2}}$ ​ + + + 5. (a) The result follows from + $+ e^z = e^{x + yi} = e^xe^{yi} = e^x(\cos(y) + i\sin(y)). +$ + + (b) $\left|e^z\right| = e^x$, $\mathrm{arg}\left(e^z\right) = y$ + + + 11. (a) For all $z$, + $+ \sin(z) = \sum_{n=0}^\infty \frac{(-1)^nz^{2n+1}}{(2n+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \frac{z^9}{9!} - \cdots . +$ + + (b) For all $z$, + $+ \cos(z) = \sum_{n=0}^\infty \frac{(-1)^nz^{2n}}{2n!} = 1 - \frac{z^2}{2} + \frac{z^4}{4!} - \frac{z^6}{6!} + \frac{z^8}{8!} - \cdots . +$ + + 15. (b) Hint: Consider what happens if $z = x + yi$ where $y$ is not between $-\pi$ and $\pi$. + + ---- + + **Section 7.3** + + 2. (a) $f'(t) = -2\sin(2t) + 2i\cos(2t)$ + + %%(c)%% $\displaystyle{g'​(t) = \frac{1}{2}\cos\left(\frac{t}{2}\right) - \frac{1}{2}i\sin\left(\frac{t}{2}\right)}$ ​ + + (d) $z'(t) = -2\mathrm{sech}(2t)\tanh(2t) + 2i\mathrm{sech}^2(2t)$ + + (e) $f'(t) = 2 + 2it$ + + (f) $g'(t) = 2t + 4it^3$ + + (g) $\displaystyle{z'​(t) = 3ie^{it}}$ ​ + + (h) $\displaystyle{h'​(t) = 3ite^{it} + 3e^{it}}$ ​ + + (i) $\displaystyle{z'​(t) = \frac{6}{t}ie^{2it} - \frac{3}{t^2}e^{2it}}$ ​ + + + 3. (a) Velocity $= -\sqrt{3} + i$ + + Speed $= 2$ + + Acceleration $= -2 - 2\sqrt{3}i$ + + %%(c)%% Velocity $= \mathrm{sech}^2(3) - i\mathrm{sech}(3)\tanh(3) = 0.009866 - 0.098837i$, + rounded to six decimal places + + Speed $= 0.099328$, rounded to six decimal places + + Acceleration $= -2\mathrm{sech}^2(3)\tanh(3) + i(\mathrm{sech}(3)\tanh^2(3) - \mathrm{sech}^3(3)) = -0.019634 + 0.097368i$, rounded to six decimal places ​ + + (e) Velocity $= -5$ + + Speed $= 5$ + + Acceleration $= -5i$ + + + 4. (a) $\displaystyle{\int_0^4 (2t + it)dt = 16 + 8i}$ + + %%(c)%% $\displaystyle{\int_0^{\frac{\pi}{2}}(-3\sin(2t) + it^3)dt = -3 + \frac{\pi^4}{64}i}$ ​ + + (e) $\displaystyle{\int_0^\pi 2te^{3it}dt = -\frac{4}{9} + \frac{2\pi}{3}i}$ ​ + + + 5. $z(t) = (1 + \sin(t)) + i(2 - \cos(t))$ + + + 6. (a) $\displaystyle{t = \frac{1}{16}s_0\sin(\alpha)}$ ​ + + (b) $\displaystyle{R = \frac{s_0^2\sin(\alpha)\cos(\alpha)}{16} = \frac{s_0^2\sin(2\alpha)}{32}}$ ​ + + + 7. (a) Maximum range $= 703.125$ feet + + (b) If $\alpha = \dfrac{\pi}{6}$, the range is $\dfrac{5625\sqrt{3}}{16}$ feet and the projectile strikes the ground after $4.6875$ seconds. + + If $\alpha = \dfrac{\pi}{3}$, the range is $\dfrac{5625\sqrt{3}}{16}$ feet and the projectile strikes the ground after $\dfrac{75\sqrt{3}}{16}$ seconds. ​ + + + 10. (b) $\displaystyle{\int_0^\infty e^{-t}dt = \varphi(0) = 1}$ + + $\displaystyle{\int_0^\infty te^{-t}dt = \frac{\varphi'​(0)}{i} = 1}$ + + $\displaystyle{\int_0^\infty t^2e^{-t}dt = \frac{\varphi''​(0)}{i^2} = 2}$ + + $\displaystyle{\int_0^\infty t^3e^{-t}dt = \frac{\varphi'''​(0)}{i^3} = 6}$ + + $\displaystyle{\int_0^\infty t^4e^{-t}dt = \frac{\varphi''''​(0)}{i^4} = 24}$ + + + 11. $\displaystyle{\varphi(\lambda) = \sqrt{2\pi}e^{-\frac{\lambda^2}{2}}}$ + + $\displaystyle{\int_{-\infty}^\infty e^{-\frac{t^2}{2}}dt = \varphi(0) = \sqrt{2\pi}}$ + + $\displaystyle{\int_{-\infty}^\infty te^{-\frac{t^2}{2}}dt = \frac{\varphi'​(0)}{i} = 0}$ + + $\displaystyle{\int_{-\infty}^\infty t^2e^{-\frac{t^2}{2}}dt = \frac{\varphi''​(0)}{i^2} = \sqrt{2\pi}}$ + + $\displaystyle{\int_{-\infty}^\infty t^3e^{-\frac{t^2}{2}}dt = \frac{\varphi'''​(0)}{i^3} = 0}$ + + $\displaystyle{\int_{-\infty}^\infty t^4e^{-\frac{t^2}{2}}dt = \frac{\varphi''''​(0)}{i^4} = 3\sqrt{2\pi}}$ + + ---- + + **Section 7.4** + + 1. (b) + + ^  Planet ​ ^  Aphelion ​ ^ + |Mercury |  0.47| + |Venus ​ |  0.73| + |Earth ​ |  1.02| + |Mars  |  1.65| + |Jupiter ​ |  5.47| + |Saturn ​ |  10.17| + |Uranus ​ |  20.22| + |Neptune ​ |  30.40| + |Pluto ​ |  46.67| + + + + 3. The distance from the sun to Halley'​s comet at aphelion is $35.2$ astronomical units. ​ + + + 5. (a) $0.9996x^2 + y^2 + 0.04x - 0.9992 = 0$ + + (b) $583$ million miles + + %%(c)%% $66,500$ miles per hour + + + 7. (a) $y_1(t) = e^t$, $y_2(t) = e^{-t}$ + + %%(c)%% $y(t) = 3e^t - e^{-t}$ + + ----