Question

求微分方程x²y’=xy—y²的通解

Answer 1

y' = y/x - (y/x)^2

设y/x = u

y = xu

y' = u + xu'

所以u + xu' = u - u^2

xu' = -u^2

- du / u^2 = dx/x

积分得

1/u = lnx + C

所以x/y = lnx + C

通解为

y = x / (lnx +C)