所谓“凑微分”是将

$$\alpha(x)f(x)+\beta(x)f'(x)$$

表示成$[G(x)f(x)]'$形式，其它项均与$f(x)$无关。例如：

$$f(x)+xf'(x)=[xf(x)]'$$

(1). 若$\beta'(x)=\alpha(x)$，则

$$\alpha(x)f(x)+\beta(x)f'(x)=[\beta(x)f(x)]'$$

(2).若$\beta'(x)\neq\alpha(x)$,设$\beta(x)\neq 0, x\in D$

$$\alpha(x)f(x)+\beta(x)f'(x)=\beta(x)\left[f'(x)+\frac{\alpha(x)}{\beta(x)}f(x)\right]$$

乘,除取值非零函数$g(x)$有

$$\frac{\beta(x)}{g(x)}\left[g(x)f'(x)+g(x)\frac{\alpha(x)}{\beta(x)}f(x)\right]$$

令$$g'(x)=g(x)\frac{\alpha(x)}{\beta(x)}$$

解得

$$g(x)=e^{\int \frac{\alpha(x)}{\beta(x)}dx}$$

我们称$g(x)$为积分因子.练习将以下个式写成全微分形式或求解常微分方程：

1. $$f(x)-xf'(x)$$

2.$$f(x) \sin x +f'(x)$$

3.$$f(x)-x^{-n}f'(x)$$

4.$$f(x)+x^{n}f'(x)$$

5.$$x^{n}f(x)+\frac{1}{1+x^{2}}f'(x)$$

6.$$\alpha(x)f(x)+\beta(x)f'(x)+h(x)=Q(x)$$

7.$$\alpha(x)f'(x)+\beta(x)f''(x)+h(x)=Q(x)$$