主题:【求教数学问题】如何判别这个函数在原点附近的性质? -- 晨枫
First consider x!=0 and y!=0
For (x,y) in a neighborhood near (0,0), we can ignore the higher order terms, and the numerator becomes 2x, denominator becomes 2y, therefore, the ordinary differential equation can be approximated as
dy/dx = x/y [1]
This gives us sign of dy/dx in four quadrants, they are positive in the first and third quadrants and negative in the second and fourth quadrants. So we know how the curves of y goes.
integrate [1] gives
y^2 = x^2 +c [2]
where c is a constant. Hence, the functions we sought is a cluster functions, we can then discuss their properties in the four quadrants.
For x=0 and y!=0, dy/dx=0, so the curve is horizontal across y axis.
For x!=0 and y=0, dy/dx is not defined, i.e. the curve is vertical across x axis, indeed, this agrees with the sign changes of dy/dx discussed above.
- 相关回复 上下关系8
🙂在这里是二元函数,不是隐函数 东海龙王 字169 2006-11-15 23:42:03
🙂dy/dx本身意味着视y为x的函数 qiaozi 字0 2006-11-15 23:44:16
🙂多元吗?那么为什么不是偏导数? qiaozi 字30 2006-11-15 22:58:32
🙂Approximation
🙂十分感谢,大体明白了 晨枫 字0 2006-11-15 23:19:18
🙂Not remember much about 1 黑洞的颜色 字523 2006-11-15 22:19:38
🙂这个主意好 晨枫 字66 2006-11-15 23:15:26
🙂高三水平??? 大大的熊 字36 2006-11-15 22:06:00