Question

微分方程，求解

dy/dx=((y-x)/(1+x))^2,

Answer 1

Answer 2

dy/dx=((y-x)/(1+x))^2
u=y-x, v=1+x
y=u+v-1
x=v-1
dy/dx=d(u+v-1)/d(v-1)
d(u+v-1)/d(v-1)=u^2/v^2
u^2dv=v^2du+v^2dv
du/dv=u^2/v^2-1
u/v=s
du=sdv+vds
s+vds/dv=s^2-1
ds/(s^2-s-1)=dv/v
ds/[(s-1/2)^2-5/4]=dlnv
(1/√5)ds/[s-1/2-√5/2)(s-1/2+√5/2)=dlnv
ln(s-1/2-√5/2)-ln(s-1/2+√5/2)=√5lnv+C0
(s-1/2-√5/2)/(s-1/2+√5/2)=Cv
v=C1[(s-1/2-√5/2)/(s-1/2+√5/2)
u=vs=C1s(s-1/2-√5/2)/(s-1/2+√5/2)

x=C1(S-1/2-√5/2)/(S-1/2+√5/2)-1
y=C1s(s-1/2-√5/2)/(s-1/2+√5/2)+C1(s-1/2-√5/2)/(s-1/2+√5/2)-1

Answer 3

令(y-x)/（1+x）=u，可转化为可分离变量的微分方程，求解即可