# 积分因子

## 方法

${\displaystyle y'+a(x)y=b(x)......(1)}$

${\displaystyle M(x)y'+M(x)a(x)y=M(x)b(x)......(2)}$

${\displaystyle (M(x)y)'=M(x)b(x)......(3)}$

${\displaystyle y(x)M(x)=\int b(x)M(x)\,dx+C,}$

${\displaystyle y(x)={\frac {\int b(x)M(x)\,dx+C}{M(x)}}.\,}$

${\displaystyle (M(x)y)'=M'(x)y+M(x)y'=M(x)b(x).\quad \quad \quad }$

${\displaystyle M'(x)=a(x)M(x)......(4)\,}$

${\displaystyle {\frac {M'(x)}{M(x)}}-a(x)=0......(5)}$

${\displaystyle M(x)=e^{\int a(x)\,dx}.}$

## 例子

${\displaystyle y'-{\frac {2y}{x}}=0.}$

${\displaystyle M(x)=e^{\int a(x)\,dx}}$
${\displaystyle M(x)=e^{\int {\frac {-2}{x}}\,dx}=e^{-2\ln x}={(e^{\ln x})}^{-2}=x^{-2}}$
${\displaystyle M(x)={\frac {1}{x^{2}}}.}$

${\displaystyle {\frac {y'}{x^{2}}}-{\frac {2y}{x^{3}}}=0}$
${\displaystyle \left({\frac {y}{x^{2}}}\right)'=0}$

${\displaystyle {\frac {y}{x^{2}}}=C}$

${\displaystyle y(x)=Cx^{2}.}$

## 一般的应用

${\displaystyle {\frac {d^{2}y}{dt^{2}}}=Ay^{2/3}}$

${\displaystyle {\frac {d^{2}y}{dt^{2}}}{\frac {dy}{dt}}=Ay^{2/3}{\frac {dy}{dt}}.}$

${\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}\right)={\frac {d}{dt}}\left(A{\frac {3}{5}}y^{5/3}\right)}$

${\displaystyle \left({\frac {dy}{dt}}\right)^{2}={\frac {6A}{5}}y^{5/3}+C_{0}}$

${\displaystyle \int {\frac {dy}{\sqrt {{\frac {6A}{5}}y^{5/3}+C_{0}}}}=t+C_{1},}$