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title: 工程數學ODE
tags: [數學]

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tags: 數學
title: 工程數學ODE
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#    <center><font color=#CCFF99>工程數學ODE</font></center>
##    <center><font color = #FFFFCC>微分</font></center>
###    <font color = #CCFFFF>常數</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color=#CCCCFF>$f(x)=3$，$f'(x)=0$</font>
:::
###    <font color = #CCFFFF>單一未知數</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color=#CCCCFF>$f(x)=x$，$f'(x)=1$</font>

<font color=#CCCCFF>$f(x)=x^2$，$f'(x)=2x$</font>

<font color=#CCCCFF>$f(x)=\sqrt{x}$，$f'(x)=\dfrac{1}{2\sqrt{x}}$</font>
:::
###    <font color = #CCFFFF>指數函數(e)</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color = #FFCC99>:sparkles:**自然常數(e)上的次方項先微分並與常數項相乘**:sparkles:</font>

<font color=#CCCCFF>$f(x)=e^x$，$f'(x)=e^x$</font>

<font color=#CCCCFF>$f(x)=e^{x^2}$，$f'(x)=2xe^{x^2}$</font>

<font color=#CCCCFF>$f(x)=e^{2x}$，$f'(x)=2e^{2x}$</font>

<font color=#CCCCFF>$f(x)=e^{2x^{2}}$，$f'(x)=4xe^{2x^2}$</font>
:::
###    <font color = #CCFFFF>對數函數(ln)</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color = #FFCC99>:sparkles:**自然對數($ln$)微分時，可以不用看常數項**:sparkles:</font>

<font color=#CCCCFF>$f(x)={\ln}(x)$，$f'(x)=\dfrac{1}{x}$</font>

<font color=#CCCCFF>$f(x)={\ln}(2x)$，$f'(x)=\dfrac{1}{x}$</font>

<font color=#CCCCFF>$f(x)={\ln}(x^2)$，$f'(x)=\dfrac{2}{x}$</font>

<font color=#CCCCFF>$f(x)={\ln}(2x^2)$，$f'(x)=\dfrac{2}{x}$</font>
:::
###    <font color = #CCFFFF>三角函數</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color=#CCCCFF>$f(x)={\sin}x$，$f'(x)={\cos}x$</font>

<font color=#CCCCFF>$f(x)={\cos}x$，$f'(x)=-{\sin}x$</font>

<font color=#CCCCFF>$f(x)=-{\sin}x$，$f'(x)=-{\cos}x$</font>

<font color=#CCCCFF>$f(x)=-{\cos}x$，$f'(x)={\sin}x$</font>
:::
###  <font color = #CCFFFF>反三角函數(arc)</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color=#CCCCFF>$f(x)={\sin}^{-1}x$，$f'(x)=\dfrac{1}{\sqrt{1-x^2}}$</font>

<font color=#CCCCFF>$f(x)={\cos}^{-1}x$，$f'(x)=\dfrac{-1}{\sqrt{1-x^2}}$</font>

<font color=#CCCCFF>$f(x)={\tan}^{-1}x$，$f'(x)=\dfrac{1}{1+x^2}$</font>

<font color=#CCCCFF>$f(x)={\sec}^{-1}x$，$f'(x)=\dfrac{1}{1+x^2}$</font>

<font color=#CCCCFF>$f(x)={\cot}^{-1}x$，$f'(x)=\dfrac{1}{|x|\sqrt{x^2-1}}$</font>

<font color=#CCCCFF>$f(x)={\csc}^{-1}x$，$f'(x)=\dfrac{-1}{|x|\sqrt{x^2-1}}$</font>
:::
##    <center><font color = #FFFFCC>積分</font></center>
###    <font color = #CCFFFF>常數</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color=#CCCCFF>$f(x)=\int3dx$，$3x=c$</font>
:::
###    <font color = #CCFFFF>單一未知數</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color = #FFCC99>**:sparkles:次方項先+1，在放至常數項的分母**:sparkles:</font>
<font color=#CCCCFF>$f(x)=\int xdx$，$\dfrac{1}{2}x^2=c$</font>

<font color=#CCCCFF>$f(x)=\int x^2dx$，$\dfrac{1}{3}x^3=c$</font>

<font color=#CCCCFF>$f(x)=\int 2x^2dx$，$\dfrac{2}{3}x^3=c$</font>

<font color=#CCCCFF>$f(x)=\int{\sqrt{x}}dx$，$\dfrac{2}{3}x^{\dfrac{3}{2}}=c$</font>
:::
###    <font color = #CCFFFF>指數函數(e)</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>

<font color=#CCCCFF>$f(x)=\int e^xdx$，$e^x=c$</font>

<font color=#CCCCFF>$f(x)=\int 2xe^xdx$，$2e^x(x-1)=c$</font> <font color=#99CC99>(分部積分)</font>

<font color=#CCCCFF>$f(x)=\int {4x^2}e^{2x}dx$，$e^{2x}(2x^2-2x+1)=c$</font> <font color=#99CC99>(分部積分)</font>

<font color=#CCCCFF>$f(x)=\int xe^{x^{2}}dx$，$\dfrac{1}{2}e^{2x}=c$</font> <font color=#99CC99>(變數變換)</font>
:::
###    <font color = #CCFFFF>對數函數(ln)</font>
:::spoiler <font color=#FFCCCC>顯示更多內容</font>
<font color = #FFCC99>:sparkles:**自然對數($ln$)積分時，80%使用分部積分**:sparkles:</font>

<font color=#CCCCFF>$f(x)=\int{\ln}(x)$，$x({\ln}x-1)=c$</font> <font color=#99CC99>(分部積分)</font>

:::
##    <center><font color = #FFFFCC>小技巧</font></center>
###    <font color = #CCFFFF><center>部分分式展開</center></font>
:::spoiler <font color=#FFCCCC>顯示範例</font>
<font color=#CCCCFF>$\dfrac{1}{x(x-2)}=0$</font>

<font color=#FFCC99>Step1．將分式看成$\dfrac{1}{x}與\dfrac{1}{x-2}$兩項</font>

<font color=#FFCC99>Step2．計算$x$項的分子時，$\dfrac{1}{x}$要為$0$，所以$x$要設為$0$，將$x=0$帶入$\dfrac{1}{x-2}$，得出$x$項分子為$\dfrac{1}{2}$</font>

<font color=#FFCC99>Step3．計算$x-2$項的分子時，$\dfrac{1}{x-2}$要為$0$，所以$x$要設為$2$，將$x=2$帶入$\dfrac{1}{x}$，得出$x-2$項分子為$\dfrac{1}{2}$</font>
:::

##    <center><font color = #FFFFCC>重點觀念</font></center>
###    <font color = #CCFFFF><center>指數函數次方相加減</center></font>
:::spoiler <font color=#FFCCCC>顯示觀念</font>
<font color=#CCCCFF>指數函數的<font color=#99CC99>次方相加</font>可以視為<font color=#99CC99>底數相乘</font></font>

<font color=#CCCCFF>例題：$e^{y^2-2x}$</font>

<font color=#FFCC99>可視為：$e^{y^2}*e^{-2x}$</font>
:::
###    <font color = #CCFFFF><center>指數函數與對數函數的關係</center></font>
:::spoiler <font color=#FFCCCC>顯示觀念</font>

<font color=#99CC99>若指數函數的次方項為對數函數則可相消</font>

<font color=#CCCCFF>例題：$e^{-2ln2}$</font>

<font color=#FFCC99>可視為：$2^{-2}$</font>
:::
###    <font color = #CCFFFF><center>對數函數基本規則</center></font>
:::spoiler <font color=#FFCCCC>顯示觀念</font>

<font color=#FFCC99>$ln(xy)=ln(x)+ln(y)$</font>

<font color=#FFCC99>$ln(\dfrac{x}{y})=ln(x)-ln(y)$</font>

<font color=#FFCC99>$ln(x^y)=yln(x)$</font>
:::
###    <font color = #CCFFFF><center>對數函數消除方法</center></font>
:::spoiler <font color=#FFCCCC>顯示觀念</font>
<font color=#CCCCFF><font color=#99CC99>同取e為底數</font>則可與ln消除</font>

<font color=#CCCCFF>例題：$ln|x|=ln|1-cos\theta|+c$</font>

<font color=#FFCC99>Step1．將ln相關的移至同側$\dfrac{ln|x|}{ln|1-cos\theta|}=c$</font>

<font color=#FFCC99>Step2．等式左右同取e為底，$\dfrac{x}{1-cos\theta}=e^c$</font>

<font color=#FFCC99>Step3．因為e為常數所以也可寫作c，所以本題答案為$\dfrac{x}{1-cos\theta}=c$</font>
:::
###    <font color = #CCFFFF><center>分子分母皆為三角函數的積分</center></font>
:::spoiler <font color=#FFCCCC>顯示觀念</font>
<font color=#CCCCFF>大多數情況使用<font color=#99CC99>變數變換</font>將分母換為u即可</font>

<font color=#CCCCFF>例題：$\int\dfrac{sin\theta}{1-cos\theta}d\theta$</font>

<font color=#FFCC99>Step1．將$1-cos\theta$設為$u$，$du=sin\theta d\theta$</font>

<font color=#FFCC99>Step2．計算$\int \dfrac{1}{u}du$，得出$ln|u|$</font>

<font color=#FFCC99>Step3．將$u$換回$(1-cos\theta)$，本題答案為$ln|1-cos\theta|$</font>
:::

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