拉普拉斯变换解微分方程
使用拉普拉斯变换解一阶、二阶微分方程。
一、核心内容
- 拉氏变换对
时域 | s域 |
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e a t \displaystyle e^{at} eat |
1 s − a \displaystyle \frac{1}{s-a} s−a1 |
1 \displaystyle 1 1 |
1 s \displaystyle \frac{1}{s} s1 |
t \displaystyle t t |
1 s 2 \displaystyle \frac{1}{s^2} s21 |
t 2 \displaystyle t^2 t2 |
2 s 3 \displaystyle \frac{2}{s^3} s32 |
- 导数的拉氏变换
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f'(x) = sF(x) - f(0) \\ f''(x) = s^2F(x) - sf(0) - f'(0)
- 留数法分解因式
二、例子
例1
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y' = 1, \qquad y(0) = 1
y′=1,y(0)=1
解:两边进行拉氏变换
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\begin{aligned} & sY(s) - y(0) = \frac{1}{s} \\ \Rightarrow &sY(s) - 1 = \frac{1}{s} \\ \Rightarrow &Y(s) = \frac{1}{s} + \frac{1}{s^2} \end{aligned}
⇒⇒sY(s)−y(0)=s1sY(s)−1=s1Y(s)=s1+s21
两边进行反拉氏变换
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y(t) = 1+t
y(t)=1+t
这个例子简单,杀鸡用了牛刀,但很有利于熟悉公式及验证算法正确性。下面看个二阶的。
例2
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y'' - y = e^{-t}, \qquad y(0) = 1, \quad y'(0)=0
y′′−y=e−t,y(0)=1,y′(0)=0
解:两边进行拉氏变换
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\begin{aligned} & s^2Y-sy(0)-y'(0) - Y= \frac{1}{1+s} \\ \Rightarrow & s^2Y - s - Y = \frac{1}{s+1} \\ \Rightarrow & Y = \frac{s^2 +s + 1}{(s+1)^2 (s-1)} \\ &\quad = \frac{A}{s+1} + \frac{B}{(s+1)^2} + \frac{C}{s-1} \end{aligned}
⇒⇒s2Y−sy(0)−y′(0)−Y=1+s1s2Y−s−Y=s+11Y=(s+1)2(s−1)s2+s+1=s+1A+(s+1)2B+s−1C
留数法分解因式
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\begin{aligned} A &= \lim_{s\to -1}\frac{d}{ds} (s+1)^2 \frac{s^2 +s + 1}{(s+1)^2 (s-1)} \\ &= \lim_{s\to -1} \frac{d}{ds} \frac{s^2 +s + 1}{s-1} \\ &= \lim_{s\to -1} \frac{(2s + 1)(s-1) - (s^2+s+1) \cdot 1}{(s-1)^2} \\ &= \frac{1}{4} \end{aligned}
A=s→−1limdsd(s+1)2(s+1)2(s−1)s2+s+1=s→−1limdsds−1s2+s+1=s→−1lim(s−1)2(2s+1)(s−1)−(s2+s+1)⋅1=41
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\begin{aligned} B &= \lim_{s\to -1} (s+1)^2 \frac{s^2 +s + 1}{(s+1)^2 (s-1)} \\ &= \lim_{s\to -1} \frac{s^2 +s + 1}{s-1} \\ &= -\frac{1}{2} \end{aligned}
B=s→−1lim(s+1)2(s+1)2(s−1)s2+s+1=s→−1lims−1s2+s+1=−21
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\begin{aligned} C &= \lim_{s\to 1} (s-1) \frac{s^2 +s + 1}{(s+1)^2 (s-1)} \\ &= \lim_{s\to 1} \frac{s^2 +s + 1}{(s+1)^2} \\ &= \frac{3}{4} \end{aligned}
C=s→1lim(s−1)(s+1)2(s−1)s2+s+1=s→1lim(s+1)2s2+s+1=43
因此
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Y =\frac{1}{4} \frac{1}{s+1} - \frac{1}{2} \frac{1}{(s+1)^2} + \frac{3}{4} \frac{1}{s-1}
Y=41s+11−21(s+1)21+43s−11
拉氏反变换得
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y(t) = \frac{1}{4} e^{-t} - \frac{1}{2} te^{-t} +\frac{3}{4} e^{t}
y(t)=41e−t−21te−t+43et
中间这一项的反变换不太懂,下次再说。