微分方程x y'' = y' ln ( y' / x )

xy''=y'ln(y'/x)
x(y''/y')=ln(y'/x)
x(lny')'=lny'-lnx
lny'=p
xp'=p-lnx
xdp=pdx-lnxdx
p/x=u
dp=xdu+udx
x^2du+xudx=xudx-lnxdx
x^2du=-lnxdx
du=-lnxdx/x^2
u=∫-lnxdx/x^2=∫lnxd(1/x)=lnx/x-∫dx/x^2=lnx/x+1/x+C0
p=lnx+1+C0x
lny'=lnx+1+C0x
y'=Ce^(lnx+1+C0x)
y=∫Ce^(lnx+1+C0x)dx
=Ce∫xe^(C0x)dx
=(Ce/C0)∫xde^(C0x)
=(Ce/C0)xe^(C0x) - (Ce/C0)∫e^(C0x)dx
=(Ce/C0)xe^(C0x) - (Ce/C0^2)e^(C0x) +C

∴微分方程xy'+p(x)y=x就是微分方程xy'+x*[(1-e^x)/(e^x)]*y∴满足条件y(ln2)=0的特解为y=-e^x+[e^(-1/2)]*[e^(e^(-x)