所谓“凑微分”是将
$$alpha(x)f(x)+eta(x)f'(x)$$
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$eta'(x)=alpha(x)$,则
$$alpha(x)f(x)+eta(x)f'(x)=[eta(x)f(x)]'$$
(2).若$eta'(x) eqalpha(x)$,设$eta(x) eq 0, xin D$
$$alpha(x)f(x)+eta(x)f'(x)=eta(x)left[f'(x)+frac{alpha(x)}{eta(x)}f(x) ight]$$
乘,除取值非零函数$g(x)$有
$$frac{eta(x)}{g(x)}left[g(x)f'(x)+g(x)frac{alpha(x)}{eta(x)}f(x) ight]$$
令$$g'(x)=g(x)frac{alpha(x)}{eta(x)}$$
解得
$$g(x)=e^{int frac{alpha(x)}{eta(x)}dx}$$
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.$$alpha(x)f(x)+eta(x)f'(x)+h(x)=Q(x)$$
7.$$alpha(x)f'(x)+eta(x)f''(x)+h(x)=Q(x)$$