拉普拉斯变换简介

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拉普拉斯变换简介

一、拉普拉斯变换的定义

定义：设函数f(x)是(0,+∞）上的分段连续函数，则其拉普拉斯变换是如下积分的结果：

0" data-eeimg="1"><span tabindex="0" class="MathJax_SVG" id="MathJax-Element-6-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='F(s)=L[f(x)]=∫0∞f(x)e−sxdx,Re(s)&gt;0'>�(�)=�[�(�)]=∫0∞�(�)�−����,��(�)>0

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-8-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='F(s)'>�(�) 称为<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-1-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='f(x)'>�(�) 的像函数，<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-10-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='f(x)'>�(�) 称为<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-14-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='F(s)'>�(�) 的原函数。

一些常用函数的拉普拉斯变换表如下：

序号	原函数	像函数
1.	1	1/s
2.	x	1/s²
3.	x²	2!/s³
4.	x³	3!/s^4
5.	e^(ax)	1/(s-a)
6.	cos(ax)	s/(s²+a²)
7.	sin(ax)	a/(s²+a²)

熟练背下这几个变换，就足可以应对绝大部分的题目了。拉普拉斯变换的作用有点像中学的对数，对数能把乘除乘方开方化成加减乘除，而拉普拉斯变换能把微积分的式子化成代数式。若把对数也当做变换的话，其过程就是对乘除的式子取对数，查对数表，运算，查反对数表得出结果；拉普拉斯变换的作用也是这样，对微积分的式子取拉普拉斯变换，查变换表，运算，查反变换得出结果。

二、举例

求函数 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-9-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='f(x)=4x2−3cos⁡2x+5e−x'>�(�)=4�2−3cos⁡2�+5�−� 的拉普拉斯变换。

利用上面的表格可得:

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-7-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='F(s)=L[f(x)]=L[4x2−3cos⁡2x+5e−x]=42!s3−3ss2+4+51s+1'>�(�)=�[�(�)]=�[4�2−3cos⁡2�+5�−�]=42!�3−3��2+4+51�+1

三、移位定理

有许多函数是由函数本身乘上一个指数函数组成的，例如： <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-13-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='e−xsin⁡2x'>�−�sin⁡2� ,我们有如下的移位定理：若已知 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-2-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='f(x)'>�(�) 的变换是 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-5-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='F(s)'>�(�) ，则 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-3-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='eaxf(x)'>����(�) 的变换是 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-11-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='F(s−a)'>�(�−�)

例如：已知 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-12-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[cos⁡2x]=ss2+4'>�[cos⁡2�]=��2+4 ,则 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-4-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[e−xcos⁡2x]=(s+1)(s+1)2+4'>�[�−�cos⁡2�]=(�+1)(�+1)2+4

四、导数的变换

为了能用于解微分方程，这条定理必须记牢。若 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-15-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[f(x)]=F(s)'>�[�(�)]=�(�) ,则导数的变换是：

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-16-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[f′(x)]=sF(s)−f(0)'>�[�′(�)]=��(�)−�(0)

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-17-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[f″(x)]=s2F(s)−sf(0)−f′(0)'>�[�″(�)]=�2�(�)−��(0)−�′(0)

......

例如：已知 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-19-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[f(x)]=ss2+9'>�[�(�)]=��2+9 , <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-18-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='f(0)=1'>�(0)=1 则 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-20-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[f′(x)]=sss2+9−1=−9s2+9'>�[�′(�)]=���2+9−1=−9�2+9

五、积分的变换

为了能用于解积分方程，这条也要牢记。若 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-21-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[f(x)]=F(s)'>�[�(�)]=�(�) ,则积分的变换是：

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-23-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[∫0xf(t)dt]=F(s)s'>�[∫0��(�)��]=�(�)�

例如：已知 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-22-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[f(x)]=ss2+9'>�[�(�)]=��2+9 ,则 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-24-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[∫0xf(t)dt]=ss2+91s=1s2+9'>�[∫0��(�)��]=��2+91�=1�2+9

六、用于求积分

由于变换本身就是个积分： <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-25-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='F(s)=∫0∞f(x)e−sxdx'>�(�)=∫0∞�(�)�−���� ,若我们令 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-26-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='s=0'>�=0 ,则有如下公式：

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-27-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='F(0)=∫0∞f(x)dx'>�(0)=∫0∞�(�)�� 即要求这个积分，只要令其拉普拉斯变换的<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-28-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='s=0'>�=0即可求出，不过要求积分本身收敛才行。举例如下：

例1.已知<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-30-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[e−xcos⁡2x]=(s+1)(s+1)2+4'>�[�−�cos⁡2�]=(�+1)(�+1)2+4，求积分： <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-29-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='I=∫0∞e−xcos⁡2xdx'>�=∫0∞�−�cos⁡2���

积分本身是存在的，所以只要令 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-31-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='s=0'>�=0即可得出：

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-32-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='I=∫0∞e−xcos⁡2xdx=(s+1)(s+1)2+4|s=0=15'>�=∫0∞�−�cos⁡2���=(�+1)(�+1)2+4|�=0=15

七、用于解微分方程

利用拉普拉斯变换的微分积分变换定理，就可以把微分方程转化成代数方程来解，下面通过举例来感受一下。

例2.解微分方程： <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-33-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='y′+y=x,y(0)=1'>�′+�=�,�(0)=1

对方程两端取拉普拉斯变换： <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-34-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='L[y′+y]=L[x]'>�[�′+�]=�[�]

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-35-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='sY(s)−f(0)+Y(s)=1s2'>��(�)−�(0)+�(�)=1�2 ，将初始条件代入，解出 <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-36-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='Y(s)'>�(�) 得：

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-37-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='Y(s)=1+s2s2(s+1)=1s2−1s+2s+1'>�(�)=1+�2�2(�+1)=1�2−1�+2�+1

再查表反变换（记熟了就直接来）即可得出微分方程的解是：

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-38-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='y=x−1+2e−x'>�=�−1+2�−�

习题：

1.求函数的拉普拉斯变换：

（a） <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-39-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='f(x)=xe−x'>�(�)=��−�

（b） <span tabindex="0" class="MathJax_SVG" id="MathJax-Element-40-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='f′(x)=1+x,f(0)=0'>�′(�)=1+�,�(0)=0

2.解微分方程：

<span tabindex="0" class="MathJax_SVG" id="MathJax-Element-41-Frame" role="presentation" style="margin: 0px; padding: 0px; border: 0px currentColor; border-image: none; text-align: left; text-transform: none; line-height: normal; text-indent: 0px; letter-spacing: normal; font-size: 16px; font-style: normal; font-weight: normal; word-spacing: normal; float: none; display: inline-block; white-space: nowrap; position: relative; direction: ltr; min-height: 0px; max-height: none; min-width: 0px; max-width: none; overflow-wrap: normal;" data-mathml='y′−2y=e−x,y(0)=1'>�′−2�=�−�,�(0)=1