Question

两种方法求解y的导数,y/x+x/y=2y

Answer 1

直接求导：（y'x-y)/x^2+(y-xy')/y^2=2y',解出y':=(y^3-x^2y)/(xy^2-x^3-2x^2y^2)

注意到 y^2+x^2=2xy^2,上式变为 y'=y(y^2-x^2)/x(y^2-x^2-(y^2+x^2))=y(2xy^2-2x^2)/(-2x^3)

=y(x-y^2)/x^2;去分母后求导更简单：2yy'+2x=2y^2+4xyy',解出y':=(x-y^2)/(2xy-y)=y(x-y^2)/x^2

因为2xy-y=x^2/y