xydy/dx=x^2+y^2
xydy=x^2dx+y^2dx
xdy^2=2x^2dx+2y^2dx
y^2=ux
dy^2=udx+xdu
x(udx+xdu)=2x^2dx+2uxdx
udx+xdu=2xdx+2udx
xdu-udx=2xdx
du/x-udx/x^2=2dx/x
d(u/x)=2dlnx
u/x=C+2lnx
u=Cx+2xlnx
y^2/x=Cx+2xlnx
通解
y^2=Cx^2+2x^2lnx
y|x=e =2e
4e^2=C4e^2+8e^2
C= -1
特解y^2=2x^2lnx-x^2