第二章

2-1 (a) $ 2 \frac{\mathrm{d}^{3}}{\mathrm{d}t^{3}} v_{\mathrm{o}}(t) + 5 \frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}} v_{\mathrm{o}}(t) + 5 \frac{\mathrm{d}}{\mathrm{d}t} v_{\mathrm{o}}(t) + 3 v_{\mathrm{o}}(t) = 2 \frac{\mathrm{d}}{\mathrm{d}t} e(t) $

(b) $ (L^{2}-M^{2})\frac{\mathrm{d}^{4}}{\mathrm{d}t^{4}}v_{\mathrm{o}}(t)+2RL\frac{\mathrm{d}^{3}}{\mathrm{d}t^{3}}v_{\mathrm{o}}(t)+\left(\frac{2L}{C}+R^{2}\right)\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}v_{\mathrm{o}}(t)+\frac{2R}{C}\frac{\mathrm{d}}{\mathrm{d}t}v_{\mathrm{o}}(t)+\frac{1}{C^{2}}v_{\mathrm{o}}(t)=MR\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}e(t) $

$$ \begin{align*}(c)&CC_{1}\frac{\mathrm{d}^{3}}{\mathrm{d}t^{3}}v_{\mathrm{o}}(t)+\left(\frac{C_{1}}{R}+\frac{C}{R_{1}}\right)\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}v_{\mathrm{o}}(t)+\left(\frac{C}{L_{1}}+\frac{1}{R_{1}R}\right)\frac{\mathrm{d}}{\mathrm{d}t}v_{\mathrm{o}}(t)+\frac{1}{RL_{1}}v_{\mathrm{o}}(t)\\&=\frac{\mu}{R_{1}}\frac{\mathrm{d}}{\mathrm{d}t}i(t)\end{align*} $$

(d) $ (1 - \mu)C \frac{\mathrm{d}}{\mathrm{d}t}v_{o}(t) + \frac{1}{R}v_{o}(t) = \frac{\mu}{R}e(t) $

$$ \begin{aligned}2-2\quad&\frac{\mathrm{d}^{3}}{\mathrm{d}t^{3}}v_{2}(t)+\frac{m_{1}f_{2}+m_{2}f_{1}}{m_{1}m_{2}}\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}v_{2}(t)+\frac{(m_{1}+m_{2})k+f_{1}f_{2}}{m_{1}m_{2}}\frac{\mathrm{d}}{\mathrm{d}t}v_{2}(t)+\frac{(f_{1}+f_{2})k}{m_{1}m_{2}}v_{2}(t)\\&=\frac{k}{m_{1}m_{2}}e(t)\end{aligned} $$

2 - 3 $ \frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}y(t) + \frac{f}{m}\frac{\mathrm{d}}{\mathrm{d}t}y(t) + \frac{k}{m}y(t) = \frac{f}{m}\frac{\mathrm{d}}{\mathrm{d}t}x(t) + \frac{k}{m}x(t) $

2-4 (1) $ \mathrm{e}^{-t}(\cos t+3\sin t) $

(2) $ (3t+1)e^{-t} $

(3) $ 1 - (t + 1)e^{-t} $

2-5 (1) $ r(0_{+})=0 $

(2) $ r(0_{+})=3 $

(3) $ r(0_{+}) = 1 $ $ r'(0_{+}) = \frac{3}{2} $

(1) $ \underbrace{4e^{-t}-3e^{-2t}}_{} $ - $ \underbrace{2e^{-t}+\frac{1}{2}e^{-2t}+\frac{3}{2}}_{} $

零输入响应 零状态响应

(2）$\underbrace{4e^{-t}-3e^{-2t}}_{\text{零输入响应}}+\underbrace{e^{-t}-e^{-2t}}_{\text{零状态响应}}$ 强迫响应等于零

$$ \begin{array}{r l}{\mathbf{2}-\mathbf{7}}&{{}v_{o}(t)=\left(E\mathrm{e}^{-\frac{t}{R C}}-R I_{\mathrm{S}}\mathrm{e}^{-\frac{t}{R C}}+R I_{\mathrm{S}}\right)u(t)}\end{array} $$

2-8 (1) $ i(0_{-}) = i(0_{+}) = 0 $, $ i'(0_{-}) = 0 $, $ i'(0_{+}) = 10 $

(2)

$$ \begin{aligned}&\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}i\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}i\left(t\right)+i\left(t\right)=0\qquad\left(t\geqslant0_{+}\right)\\&i\left(t\right)=\frac{20}{\sqrt{3}}\mathrm{e}^{-\frac{1}{2}t}\sin\left(\frac{\sqrt{3}}{2}\ t\right)\\ \end{aligned} $$

(3) $ \frac{d^{2}}{dt^{2}}i(t)+\frac{d}{dt}i(t)+i(t)=\frac{d}{dt}e(t) $，其中 $ e(t)=10+10u(t) $

2-9 (1) $ h(t)=2\delta(t)-6\mathrm{e}^{-3t}u(t) $

$$ g(t)=2\mathrm{e}^{-3t}u(t) $$

(2)

$$ h\left(t\right)=\mathrm{e}^{-\frac{1}{2}t}\left[\cos\left(\frac{\sqrt{3}}{2}t\right)+\frac{1}{\sqrt{3}}\sin\left(\frac{\sqrt{3}}{2}t\right)\right]u\left(t\right) $$

$$ g\left(t\right)=\left\{\mathrm{e}^{-\frac{1}{2}t}\left[-\cos\left(\frac{\sqrt{3}}{2}t\right)+\frac{1}{\sqrt{3}}\mathrm{s i n}\left(\frac{\sqrt{3}}{2}t\right)\right]+1\right\}u\left(t\right) $$

(3)

$$ h(t)=\mathrm{e}^{-2t}u(t)+\delta(t)+\delta^{\prime}(t) $$

$$ g(t)=\delta(t)+\bigg(\frac{3}{2}-\frac{1}{2}\mathrm{e}^{-2t}\bigg)u(t) $$

$$ \mathbf{2-10}\quad h\left(t\right)=\left(\frac{1}{4}\mathrm{e}^{-t}+\frac{7}{4}\mathrm{e}^{-5t}\right)u\left(t\right) $$

$$ \mathbf{2}-\mathbf{11}\quad r\left(0_{-}\right)=\frac{1}{2},r^{\prime}\left(0_{-}\right)=-\frac{1}{2},C=\frac{1}{2} $$

2-12 (1) $ r_{zi}(t)=\mathrm{e}^{-t} $ (当 $ t \geqslant 0 $)

(2)

$$ r_{3}(t)=(2-t)\mathrm{e}^{-t}u(t) $$

2-13

(1)

$$ \frac{1}{\alpha}(1-\mathrm{e}^{-\alpha t})u(t) $$

(2)

$$ \cos(\omega t+45^{\circ}) $$

(3)

$$ \left\{\begin{aligned}{}&{{}0}&{(t<1,t>3)}\\ {}&{{}\frac{1}{2}(t^{2}-1)}&{(1

(4)

$$ \cos[\omega(t+1)]-\cos[\omega(t-1)] $$

(5)

$$ \frac{a\sin t-\cos t+e^{-at}}{a^{2}+1}u(t) $$

2-16

$$ A=\frac{1}{1-\mathrm{e}^{-3}} $$

2-17

$$ h(t)=\mathrm{e}^{t-1}u(3-t) $$

2-18

$$ \frac{1}{2}\mathrm{e}^{-2\prime}u\left(t\right) $$

2-19 (b) $ u(-t) + (2 - e^{-t})u(t) $

$$ \begin{aligned}{}&{{}\operatorname{(c)}\left\{\begin{aligned}{}&{{}2(1-\cost)}&{(0

(d)

$$ \left\{\begin{aligned}&\frac{1}{2}t^{2}\quad(0

对于 $ n < t < n + 1 $ （ $ n \geqslant 2 $）

$$ \left(-1\right)^{n}\left[2\left(t-\frac{2n+1}{2}\right)^{2}-\frac{1}{2}\right] $$

(e) $ 1 - \cos(t - 1) $ (t > 1)

(f)

$$ \frac{1}{\pi}\left[1-\cos(\pi t)\right]\left[u\left(t\right)-u\left(t-2\right)\right]*\sum_{k=0}^{\infty}\delta\left(t-3k\right) $$

2-20 $ u(t)-u(t-1) $

2-21

$$ \left\{\begin{array}{l l}{\mathrm{e}^{-t}-\mathrm{e}^{-2t}}&{(02)}\end{array}\right. $$

(2)

$$ \beta=-\mathrm{e}^{-4}\int_{0}^{2}\mathrm{e}^{2\tau}x\left(\tau\right)\mathrm{d}\tau $$

2-23

(1)

$$ \frac{1}{2}\left[\delta\left(t+\frac{1}{2}\right)+\delta\left(t-\frac{1}{2}\right)\right] $$

(2)

$$ \sum_{k=-\infty}^{\infty}\delta(t-k\pi) $$

2-24

$$ \sum_{k=-\infty}^{\infty}(-1)^{k}\left[u\left(t+\frac{\pi}{2}-k\pi\right)-u\left(t-\frac{\pi}{2}-k\pi\right)\right] $$

2-27 (1) $ A e^{-at} u(t) $

(2)

$$ A t\mathrm{e}^{-a t}u(t) $$

(3)

$$ \frac{A}{\alpha-\beta}(\mathrm{e}^{-\beta t}-\mathrm{e}^{-\alpha t})u(t) $$