解微分方程dx/dy=－x－y^2
问题描述：
解微分方程dx/dy=－x－y^2
答
x+y^2=u
x=u-y^2
dx=du-2ydy
dx/dy=du-2y
du/dy=2y-u
2y-u=v
du=2dy-dv
2-dv/dy=v
dv/dy=2-v
dv/(2-v)=dy
ln(v-2)= -y+c0
v-2=Ce^(-y)
v=2+Ce^(-y)
u=2y-v=2y-2-Ce^(-y)
x=u-y^2=2y-y^2-2-C1e^(-y)
答
x'+x=-y²(e^y)(x'+x)=-y²e^y[xe^y]'=-y²e^yxe^y=∫-y²e^ydy=-y²e^y+∫2ye^y=-y²e^y+2ye^y-∫2e^ydy=-y²e^y+2ye^y-2e^y+Cx=-y²+2y-2+Ce^(-y)