积分因子

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

方法

$y'+a(x)y = b(x)......(1)$

$M(x)y' + M(x)a(x)y = M(x)b(x)......(2)$

$(M(x)y)' = M(x)b(x)......(3)$

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

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

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

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

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

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

例子

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

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

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

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

$y(x) = Cx^2.$

一般的应用

$\frac{d^2 y}{d t^2} = A y^{2/3}$

$\frac{d^2 y}{d t^2} \frac{d y}{d t} = A y^{2/3} \frac{d y}{d t}.$

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

$\left(\frac{d y}{d t}\right)^2 = \frac{6 A}{5} y^{5/3} + C_0$

$\int \frac{d y}{\sqrt{\frac{6 A}{5} y^{5/3} + C_0}} = t + C_1,$

参考文献

• Adams, R. A. Calculus: A Complete Course, 4th ed. Reading, MA: Addison Wesley, 1999.