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.