全微分方程(x^2-y)dx+axdy=0(a为常数)的通解为
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dy/dx=(x^2-y)/ax
(x^2-y)/ax=2u
y=x^2-2axu
dy=2xdx-2adu-2axdu
2x-2(a-ax)du/dx=2u
x-u=(a-ax)du/dx
(x-u)/(a-ax)=du/dx
(x-u)/(1-x)=v
u=x-v+xv
du=dx-dv+vdx+xdv
v/a=(1+v)+(x-1)dv/dx
(v/a-1-v)=(x-1)dv/dx
dx/(x-1)=dv/[v(1/a-1)-1)]
ln(x-1)=[a/(1-a)]ln(v(1/a-1)-1]+C0
C1(x-1)^(1/a-1)=v(1/a-1)-1
v=C2(x-1)^(1/a-1)+a/(1-a)
u=x-v+xv
=x-C2(x-1)^(1/a-1)-a/(1-a)+C2x^(1/a)
y=x^2-2axu
=x^2-2ax^2+C3(x-1)^(1/a)-[2a^2/(1-a)]x-C3x^(1/a+1)
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