In[]:=
$Version
Out[]=
14.0.0 for Microsoft Windows (64-bit) (December 13, 2023)
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[]=
DSolve-(-2+v)
W[ξ]
-v+
v1
μ
(-
3/2
W[ξ]
+
5/2
W[ξ]
)+k-
2
′
W
[ξ]
4
3/2
W[ξ]
+
′′
W
[ξ]
2
W[ξ]
0,W,ξ
In[]:=
B1=Collect(A1[[1,1]])*
W[ξ]
//FullSimplify,W[ξ](*Solutionone:*)
Out[]=
(2-v)W[ξ]+v+
v1
μ
2
W[ξ]
-
(v1+vμ)
3
W[ξ]
μ
-
k
2
′
W
[ξ]
4W[ξ]
+
1
2
k
′′
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[]=
DSolve(-2+v)
W[ξ]
-v+
v1
μ
(
3/2
W[ξ]
-
5/2
W[ξ]
)+k
2
′
W
[ξ]
4
3/2
W[ξ]
-
′′
W
[ξ]
2
W[ξ]
0,W,ξ
In[]:=
B2=Collect(A2[[1,1]])*
W[ξ]
//FullSimplify,W[ξ](*Solutiontwo:*)
Out[]=
(-2+v)W[ξ]-
(v1+vμ)
2
W[ξ]
μ
+v+
v1
μ
3
W[ξ]
+
k
2
′
W
[ξ]
4W[ξ]
-
1
2
k
′′
W
[ξ]