解答题
7.设P(x)在[0，+∞)连续且为负值，y=y(x)在[0，+∞)连续，在(0，+∞)满足)y'+P(x)y＞0且y(0)≥0，求证：y(x)在[0，+∞)单调增加．
【正确答案】由y'+P(x)y＞0(x＞0)
y(x)]'＞0 (x＞0)，又
y(x)在[0，+∞)连续，
y(x)＞0(x≥0)
y'(x)＞－P(x)y(x)＞0 (x＞0)
y(x)在[0，+∞)单调增加．
y(x)]'＞0 (x＞0)，又y(x)在[0，+∞)连续，y(x)＞0(x≥0)y'(x)＞－P(x)y(x)＞0 (x＞0)y(x)在[0，+∞)单调增加．【答案解析】