Question

求解下列初值问题

(y-2xy)dx+x^2dy=0，y|x=1=e
y'-y=xy^5,y(0)=1

Answer 1

令：v=y/x,y=xv,dy=vdx+xdv
dy/键燃仿dx = -(y^2-2xy)/x^2
(vdx+xdv)/dx = 2v - v^2
v+xdv/dx = 2v - v^2
xdv/dx = v - v^2
dv/[v(1 - v)] = dx/x
∫dv/[v(1 - v)] = ∫稿纤dx/x
∫段戚dv/v + ∫dv/(1 - v)] = ∫dx/x
lnv - ln(1-v) = lnx
ln(v/(1-v))=lnx+lnc
v/(1-v)=cx
(y/x)/(1-(y/x))=cx
y/(x-y)=cx
y=cx(x-y)