Question

方程ydx-xdy=(x^2+y^2)dx的通解

Answer 1

y dx - x dy = (x² + y²) dx

y - x • dy/dx = x² + y²

y' = y/x - y²/x - x

(令y = - xv,y' = - (xv' + v) = - xv' - v)

- xv' - v = - v - xv² - x

- v' = - v² - 1

dv/(v² + 1) = 1

arctan(v) = x + C

v = tan(x + C)

- y/x = tan(x + C)

y = - xtan(x + C)