【补充习题四】凑微分技巧与积分因子法解常微分方程
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.$$\alpha(x)f'(x)+\beta(x)f''(x)+h(x)=Q(x)$$
