In[]:=
$Version
Out[]=
12.0.0 for Microsoft Windows (64-bit) (April 6, 2019)
In[]:=
A1=DSolveChangeVariablesDSolveu''[x]-(v-2)*u[x]-v+
v1
μ
*(-u[x]^3+u[x]^5)0,u,x,W,ξ,u[x]W[ξ]^(1/2),xct+
ξ
k

Out[]=
DSolveChangeVariablesDSolve-(-2+v)u[x]-v+
v1
μ
(-
3
u[x]
+
5
u[x]
)+
′′
u
[x]0,u,x,W,ξ,u[x]
W[ξ]
,xct+
ξ
k

In[]:=
B1=Collect(A1[[1,1]])*
W[ξ]
//FullSimplify,W[ξ](*Solutionone:*)
Out[]=
′′
u
[x]u[x]-2+v+v+
v1
μ
2
u[x]
(-1+
2
u[x]
)
W[ξ]
In[]:=
A2=DSolveChangeVariablesDSolveu''[x]-(v-2)*u[x]-v+
v1
μ
*(-u[x]^3+u[x]^5)0,u,x,W,ξ,u[x]-W[ξ]^(1/2),xct+
ξ
k

Out[]=
DSolveChangeVariablesDSolve-(-2+v)u[x]-v+
v1
μ
(-
3
u[x]
+
5
u[x]
)+
′′
u
[x]0,u,x,W,ξ,u[x]-
W[ξ]
,xct+
ξ
k

In[]:=
B2=Collect(A2[[1,1]])*
W[ξ]
//FullSimplify,W[ξ](*Solutiontwo:*)
Out[]=
′′
u
[x]u[x]-2+v+v+
v1
μ
2
u[x]
(-1+
2
u[x]
)
W[ξ]