Question

求大神帮忙写出化简常微分方程由倒数第二步到最后一步的具体步骤！

Answer 1

由y=ux得，u=y/x，代入
ln|(1-y/x)/(1+y/x)|-1/2*ln|y²/x²-1|=ln|x|+c
ln|(x-y)/(x+y)|-1/2*ln|(y+x)(y-x)/x²|=ln|x|+c
ln|x-y|-ln|x+y|-1/2*[ln|y+x|+ln|y-x|-lnx²]=ln|x|+c
ln|x-y|-ln|x+y|-1/2*ln|y+x|-1/2*ln|y-x|+ln|x|=ln|x|+c
1/2*ln|y-x|-3/2*ln|y+x|=c
ln|y-x|-3ln|y+x|=2c
ln|y-x|-ln|y+x|^3=2c
ln|(y-x)/(y+x)^3|=2c
两边同时取指数，则
(y-x)/(y+x)^3=±e^2c
令c1=±e^2c，则有y-x=c1*(y+x)^3
再将c1换成c，则有y-x=c*(y+x)^3

Answer 2

[(1-u)/(1+u)]^2/[(u^2-1)x^2]=2C
(u-1)/[(1+u)^3x^2]=2C
(y/x-1)/[(1+y/x)^3x^2]=2C
(y-x)/(x+y)^3=2C=C1

Answer 3