微分方程x*dy/dx+y=3它的通解是Y=C/x+3吗,我把通解带进去求等是不相等,xdy+(y-3)dx=0到xd(y-3)+(y-3)dx=0不懂，xd(y-3)怎么变过去的呢

Question

微分方程x*dy/dx+y=3
它的通解是Y=C/x+3吗,我把通解带进去求等是不相等,
xdy + (y-3)dx = 0
到xd(y-3) + (y-3)dx = 0不懂，xd(y-3) 怎么变过去的呢

Answer 1

x*dy/dx+y=3
两边同时乘以dx
xdy + (y-3)dx = 0
so
xd(y-3) + (y-3)dx = 0
so
d[x(y-3)] = 0
so
x(y-3)=C C is a constant
so
y=C/x+3
检验： x*dy/dx+y=x*(-C/x^2) + C/x+3 = -C/x + C/x+3 = 3, 满足原方程.