(x^3+xy^2)dx+(x^2y+y^3)dy=0的通解,具体的解决步骤,一定要具体

Question

(x^3+xy^2)dx+(x^2y+y^3)dy=0的通解,具体的解决步骤,一定要具体

Answer 1

方程分离变量后两边积分得1/2ln(1+y^2)=1/2ln 1-x^2的绝对值+ln c1的绝对值.
(x3 + xy2)dx + (x2y + y3)dy = 0
x(x2 + y2) + y(x2 + y2)dy/dx = 0
(x2 + y2)(x + ydy/dx) = 0
x2 + y2 = 0 或 x + ydy/dx = 0
y dy = - x dx
y2/2 = - x2/2 + C
y2 = - x2 + C
x2 + y2 = C
所以通解为x2 + y2 = 0 或 x2 + y2 = C
合并就是x2 + y2 = C