In[]:=
dG=
′
G
;​​subst1=ΞdG[ξ_]dG[ξ]/G[ξ]^2;​​subst2=Ξd
′
G
[ξ_]A+BΞdG[ξ]^2+cΞdG[ξ];​​subst3=
′
′
ΞdG
[ξ_]D[
′
ΞdG
[ξ]/.subst2,ξ]/.subst2/.subst1;​​​​a={a0,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20};​​​​​​U[ξ_]:=Sum[a[[i+2]]*(H+ΞdG[ξ])^i,{i,p,q}]​​p=-1;q=1;​​U[ξ]​​eq1=2*(ν^2-k^2)*D[U[ξ],ξ]+2(αk^2ν^2)*D[U[ξ],{ξ,2}]-3νk^2*D[U[ξ],ξ]^2/.subst1/.subst2/.subst3//Expand​​​​
Out[]=
a1+
a0
H+ΞdG[ξ]
+a2(H+ΞdG[ξ])
Out[]=
-2a2
2
k
′
ΞdG
[ξ]+2a2
2
ν
′
ΞdG
[ξ]+
2a0
2
k
′
ΞdG
[ξ]
2
H+
′
G
[ξ]
2
G[ξ]
-
2a0
2
ν
′
ΞdG
[ξ]
2
H+
′
G
[ξ]
2
G[ξ]
-3
2
a2
2
k
ν
2
′
ΞdG
[ξ]
-
3
2
a0
2
k
ν
2
′
ΞdG
[ξ]
4
H+
′
G
[ξ]
2
G[ξ]
+
4a0
2
k
α
2
ν
2
′
ΞdG
[ξ]
3
H+
′
G
[ξ]
2
G[ξ]
+
6a0a2
2
k
ν
2
′
ΞdG
[ξ]
2
H+
′
G
[ξ]
2
G[ξ]
+2a2
2
k
α
2
ν
′′
ΞdG
[ξ]-
2a0
2
k
α
2
ν
′′
ΞdG
[ξ]
2
H+
′
G
[ξ]
2
G[ξ]