常微分方程【大学常用公式】

伯努利方程:

\begin{array}{*{20}{l}}
{\text{形}\text{如}}\\
{\frac{{ \text{d} y}}{{ \text{d} x}}+p{ \left( {x} \right) }y=f{ \left( {x} \right) }\mathop{{y}}\nolimits^{{n}}{ \left( {n \neq 0,1} \right) }}\\
{\text{称}\text{为}\text{伯}\text{努}\text{利}\text{方}\text{程}\text{,}\text{是}\text{一}\text{种}\text{非}\text{线}\text{性}\text{的}\text{一}\text{阶}\text{微}\text{分}\text{方}\text{程}}\\
{\text{将}\text{伯}\text{努}\text{利}\text{方}\text{程}\text{两}\text{端}\text{除}\text{以}\mathop{{y}}\nolimits^{{n}}\text{,}\text{得}}\\
{\mathop{{y}}\nolimits^{{-n}}\frac{{ \text{d} y}}{{ \text{d} x}}+p{ \left( {x} \right) }\mathop{{y}}\nolimits^{{1-n}}=f{ \left( {x} \right) }}\\
{\text{令}z=\mathop{{y}}\nolimits^{{1-n}}\text{有}}\\
{\frac{{ \text{d} z}}{{ \text{d} x}}={ \left( {1-n} \right) }\mathop{{y}}\nolimits^{{-n}}\frac{{ \text{d} y}}{{ \text{d} x}}}\\
{\frac{{1}}{{1-n}}\frac{{ \text{d} z}}{{ \text{d} x}}+p{ \left( {x} \right) }z=f{ \left( {x} \right) }}\\
{\text{则}\text{原}\text{方}\text{程}\text{化}\text{为}\text{线}\text{性}\text{方}\text{程}\text{进}\text{行}\text{求}\text{解}}
\end{array}

 

初值问题:

$$\left\{ \begin{array}{*{20}{l}}
{\mathop{{y}}\nolimits^{{ \left( {n} \right) }}=f{ \left( {x,y,{y \prime }, \cdots ,\mathop{{y}}\nolimits^{{ \left( {n-1} \right) }}} \right) }}\\
{y{ \left( {\mathop{{x}}\nolimits_{{0}}} \right) }=\mathop{{y}}\nolimits_{{0}}}\\
{y \prime { \left( {\mathop{{x}}\nolimits_{{0}}} \right) }={\mathop{{y}}\nolimits_{{0}} \prime }}\\
{ \cdots }\\
{\mathop{{y}}\nolimits^{{ \left( {n-1} \right) }}{ \left( {\mathop{{x}}\nolimits_{{0}}} \right) }=\mathop{{\mathop{{y}}\nolimits_{{0}}}}\nolimits^{{ \left( {n-1} \right) }}}
\end{array}\right.$$

 

第一类可化为分离变量微分方程:

\begin{array}{*{20}{l}}
{\text{若}\text{一}\text{阶}\text{显}\text{示}\text{方}\text{程}}\\
{\frac{{ \text{d} y}}{{ \text{d} x}}=f{ \left( {x,y} \right) }}\\
{\text{右}\text{端}f{ \left( {x,y} \right) }\text{可}\text{以}\text{改}\text{写}\text{为}\frac{{y}}{{x}}\text{的}\text{函}\text{数}g{ \left( {\frac{{y}}{{x}}} \right) }\text{,}\text{即}\text{有}}\\
{\frac{{ \text{d} y}}{{ \text{d} x}}=g{ \left( {x,y} \right) }}\\
{\text{则}\text{称}\text{该}\text{方}\text{程}\text{为}\text{一}\text{阶}\text{齐}\text{次}\text{微}\text{分}\text{方}\text{程}\text{,}\text{可}\text{化}\text{为}\text{变}\text{量}\text{可}\text{分}\text{离}\text{方}\text{程}\text{进}\text{行}\text{求}\text{解}}\\
{\text{令}u=\frac{{y}}{{x}}\text{,}\text{方}\text{程}\text{通}\text{解}\text{为}}\\
{Cx=\mathop{{e}}\nolimits^{{{}_{ }^{ } \int _{ }^{ }\frac{{ \text{d} u}}{{g{ \left( {u} \right) }-u}}}}}
\end{array}

 

第二类可化为分离变量微分方程:

\begin{array}{*{20}{l}}
{\text{形}\text{如}}\\
{\frac{{ \text{d} y}}{{ \text{d} x}}=f{ \left( {\frac{{\mathop{{a}}\nolimits_{{1}}x+\mathop{{b}}\nolimits_{{1}}y+\mathop{{c}}\nolimits_{{1}}}}{{\mathop{{a}}\nolimits_{{2}}x+\mathop{{b}}\nolimits_{{2}}y+\mathop{{c}}\nolimits_{{2}}}}} \right) }{ \left( {\mathop{{\mathop{{c}}\nolimits_{{1}}}}\nolimits^{{2}}+\mathop{{\mathop{{c}}\nolimits_{{2}}}}\nolimits^{{2}} \neq 0} \right) }}\\
{\text{做}\text{变}\text{量}\text{替}\text{换}{ \left\{ {\begin{array}{*{20}{c}}
{x= \xi + \alpha }\\
{y= \eta + \beta }
\end{array}}\right. },\text{其}\text{中} \alpha \text{和} \beta \text{为}\text{待}\text{定}\text{常}\text{数}\text{,}\text{其}\text{取}\text{值}\text{满}\text{足}}\\
{ \left\{ {\begin{array}{*{20}{c}}
{\mathop{{a}}\nolimits_{{1}} \alpha +\mathop{{b}}\nolimits_{{1}} \beta +\mathop{{c}}\nolimits_{{1}}=0}\\
{\mathop{{a}}\nolimits_{{2}} \alpha +\mathop{{b}}\nolimits_{{2}} \beta +\mathop{{c}}\nolimits_{{2}}=0}
\end{array}}\right. }\\
{\text{原}\text{方}\text{程}\text{可}\text{化}\text{为}\text{第}\text{一}\text{类}\text{可}\text{化}\text{为}\text{的}\text{变}\text{量}\text{分}\text{离}\text{微}\text{分}\text{方}\text{程}}\\
{\frac{{ \text{d} \eta }}{{ \text{d} \xi }}=f{ \left( {\frac{{\mathop{{a}}\nolimits_{{1}} \xi +\mathop{{b}}\nolimits_{{1}} \eta }}{{\mathop{{a}}\nolimits_{{2}} \xi +\mathop{{b}}\nolimits_{{2}} \eta }}} \right) }}
\end{array}

 

二阶常系数非齐次线性微分方程:

\begin{array}{*{20}{l}}
{\text{对}\text{二}\text{阶}\text{方}\text{程}}\\
{y ” +p{y \prime }+qy=f{ \left( {x} \right) }}\\
{\text{先}\text{求}\text{对}\text{应}\text{齐}\text{次}\text{方}\text{程}}\\
{y ” +p{y \prime }+qy=0}\\
{\text{的}\text{通}\text{解}y\text{,}\text{再}\text{根}\text{据}f{ \left( {x} \right) }\text{求}\text{另}\text{一}\text{个}\text{特}\text{解}\mathop{{y}}\nolimits_{{0}}}
\end{array}

 

全微分方程:

\begin{array}{*{20}{l}}
{\text{若}\text{方}\text{程}}\\
{P{ \left( {x,y} \right) } \text{d} x+Q{ \left( {x,y} \right) } \text{d} y=0}\\
{\text{的}\text{左}\text{端}\text{是}\text{某}\text{函}\text{数}\text{的}\text{全}\text{微}\text{分}\text{方}\text{程}\text{,}\text{即}}\\
{ \text{d} u{ \left( {x,y} \right) }=P{ \left( {x,y} \right) } \text{d} x+Q{ \left( {x,y} \right) } \text{d} y=0}\\
{\text{其}\text{中}}\\
{ \left\{ \begin{array}{*{20}{l}}
{\frac{{ \partial u}}{{ \partial x}}=P{ \left( {x,y} \right) }}\\
{\frac{{ \partial u}}{{ \partial x}}=Q{ \left( {x,y} \right) }}
\end{array}\right. }\\
{\text{则}}\\
{u{ \left( {x,y} \right) }=C}\\
{\text{是}\text{原}\text{方}\text{程}\text{的}\text{通}\text{解}}
\end{array}

 

微分形式变量可分离方程:

\begin{array}{*{20}{l}}
{\mathop{{M}}\nolimits_{{1}}{ \left( {x} \right) }\mathop{{N}}\nolimits_{{1}}{ \left( {y} \right) } \text{d} x=\mathop{{M}}\nolimits_{{2}}{ \left( {x} \right) }\mathop{{N}}\nolimits_{{2}}{ \left( {y} \right) } \text{d} y}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{\mathop{{N}}\nolimits_{{2}}{ \left( {y} \right) }}}{{\mathop{{N}}\nolimits_{{1}}{ \left( {y} \right) }}} \text{d} y={}_{ }^{ } \int _{ }^{ }\frac{{\mathop{{M}}\nolimits_{{2}}{ \left( {x} \right) }}}{{\mathop{{M}}\nolimits_{{1}}{ \left( {x} \right) }}} \text{d} x}
\end{array}

 

微分形式一阶方程:

\begin{array}{*{20}{l}}
{M{ \left( {x,y} \right) } \text{d} x+N{ \left( {x,y} \right) } \text{d} y=0}\\
{y= \varphi { \left( {x,\mathop{{C}}\nolimits_{{1}},\mathop{{C}}\nolimits_{{2}}, \cdots ,\mathop{{C}}\nolimits_{{n}}} \right) }}
\end{array}

 

显示变量可分离方程:

\begin{array}{*{20}{l}}
{\frac{{ \text{d} y}}{{ \text{d} x}}=f{ \left( {x} \right) }g{ \left( {y} \right) }}\\
{{}_{ }^{ } \int _{ }^{ }\frac{{ \text{d} y}}{{g{ \left( {y} \right) }}}={}_{ }^{ } \int _{ }^{ }f{ \left( {x} \right) } \text{d} x+C}
\end{array}

 

显示方程:

\begin{array}{*{20}{l}}
{y \prime =f{ \left( {x,y} \right) }}\\
{\mathop{{y}}\nolimits^{{ \left( {n} \right) }}=f{ \left( {x,y,y, \cdots ,\mathop{{y}}\nolimits^{{ \left( {n-1} \right) }}} \right) }}
\end{array}

 

一阶线性非齐次微分方程:

\begin{array}{*{20}{l}}
{\begin{array}{*{20}{l}}
{\frac{{ \text{d} y}}{{ \text{d} x}}+p{ \left( {x} \right) }y=f{ \left( {x} \right) }}\\
{u={}_{ }^{ } \int _{ }^{ }p{ \left( {x} \right) } \text{d} x}
\end{array}}\\
{y=C\mathop{{e}}\nolimits^{{-u}}+\mathop{{e}}\nolimits^{{-u}}{}_{ }^{ } \int _{ }^{ }f{ \left( {x} \right) }\mathop{{e}}\nolimits^{{u}} \text{d} x}
\end{array}

 

一阶线性齐次微分方程:

\begin{array}{*{20}{l}}
{\frac{{ \text{d} y}}{{ \text{d} x}}+p{ \left( {x} \right) }y=0}\\
{y=C\mathop{{e}}\nolimits^{{-{}_{ }^{ } \int _{ }^{ }p{ \left( {x} \right) } \text{d} x}}}
\end{array}

 

隐式方程:

\begin{array}{*{20}{l}}
{F{ \left( {x,y,{y \prime }} \right) }=0}\\
{F{ \left( {x,y,{y \prime },{y ” }, \cdots ,\mathop{{y}}\nolimits^{{ \left( {n} \right) }}} \right) }=0}
\end{array}

 

二阶常系数齐次线性微分方程:

\begin{array}{*{20}{l}}
{\text{对}\text{二}\text{阶}\text{方}\text{程}}\\
{y ” +p{y \prime }+qy=0}\\
{\text{其}\text{中}p,q\text{为}\text{常}\text{数}}\\
{\text{观}\text{察}\text{其}\text{特}\text{征}\text{方}\text{程}\mathop{{r}}\nolimits^{{2}}+pr+q=0\text{的}\text{根}\mathop{{r}}\nolimits_{{1}}\text{和}\mathop{{r}}\nolimits_{{1}}\text{,}\text{其}\text{通}\text{解}\text{对}\text{照}\text{如}\text{下}}\\
{y={ \left\{ {\begin{array}{*{20}{l}}
{\mathop{{C}}\nolimits_{{1}}\mathop{{e}}\nolimits^{{\mathop{{r}}\nolimits_{{1}}x}}+\mathop{{C}}\nolimits_{{2}}\mathop{{e}}\nolimits^{{\mathop{{r}}\nolimits_{{2}}x}}}&{\mathop{{p}}\nolimits^{{2}}-4q > 0}\\
{ \left( {\mathop{{C}}\nolimits_{{1}}+\mathop{{C}}\nolimits_{{2}}} \left) \mathop{{e}}\nolimits^{{\mathop{{r}}\nolimits_{{1}}x}}\right. \right. }&{\mathop{{p}}\nolimits^{{2}}-4q=0}\\
{\mathop{{e}}\nolimits^{{ \alpha x}}{ \left( {\mathop{{C}}\nolimits_{{1}} \text{cos} \beta x+\mathop{{C}}\nolimits_{{2}} \text{sin} \beta x} \right) }}&{\mathop{{p}}\nolimits^{{2}}-4q < 0}
\end{array}}\right. }}\\
{\text{其}\text{中}\mathop{{r}}\nolimits_{{1,2}}= \alpha \pm i \beta \text{为}\text{特}\text{征}\text{方}\text{程}\text{的}\text{一}\text{对}\text{共}\text{轭}\text{复}\text{根}}\\
{ \alpha =-\frac{{p}}{{2}}}\\
{ \beta =\frac{{\sqrt{{4q-\mathop{{p}}\nolimits^{{2}}}}}}{{2}}}
\end{array}

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