高等数学微分方程求通解

Question

求过程

Answer 1

是齐次方程，令 y = xu，则 微分方程化为
u + xdu/dx = (1+u)/(1-u)
xdu/dx = (1+u)/(1-u) - u = (1+u^2)/(1-u)
(1-u)du/(1+u^2) = dx/x
arctanu - (1/2)ln(1+u^2) = lnx + lnC
e^(arctanu) = Cx√(1+u^2)
通解是 e^[arctan(y/x)] = C√(x^2+y^2)