Question

微分方程xy（dx-dy）=y^2dx+x^2dy是可分变量方程还是一介齐次还是一介线性还是全微分方程,为什么呢?

Answer 1

∵xy(dx-dy)=y²dx+x²dy ==>xydx-xydy=y²dx+x²dy

==>xydx-y²dx=x²dy+xydy

==>y(x-y)dx=x(x+y)dy

==>dy/dx=y(x-y)/[x(x+y)]

==>y'=(y/x)(1-y/x)/(1+y/x)

==>y'=f(y/x) (其中f(u)=u(1-u)/(1+u))

∴根据齐次微分方程的定义知,xy(dx-dy)=y²dx+x²dy 是齐次微分方程.