Question

微分方程（2xsiny+3x2y）dx+（x3+x2cosy+y2）dy=0的通解是______

微分方程（2xsiny+3x2y）dx+（x3+x2cosy+y2）dy=0的通解是______．

Answer 1

∵（2xsiny+3x²y）dx+（x³+x²cosy+y²）dy中的P（x，y）=2xsiny+3x²y，Q（x，y）=x³+x²cosy+y²
∴

?P

?y

＝

?Q

?x

＝3x2+2xcosy
∴u(x，y)＝

∫	(x，y) (0，0)

(2xsiny+3x2y)dx+(x3+x2cosy+y2)dy
=

∫	(x，0) (0，0)

(2xsiny+3x2y)dx+(x3+x2cosy+y2)dy+

∫	(x，y) (x，0)

(2xsiny+3x2y)dx+(x3+x2cosy+y2)dy
=

∫

x 0

0dx+

∫

y 0

(x3+x2cosy+y2)dy
=x3y+x2siny+

1

3

y3＝C，其中C为任意常数．