In[]:=
ClearAll["Global`*"]​​Clear[f]​​(*Definitionofthefirstequationه*)​​eq1=D[f[ξ],{ξ,2}]-(c/(
2
c
-1))*D[f[ξ],{ξ,1}]+(α/(
2
c
-1))*f[ξ]+β(
2
c
-1)*
3
f[ξ]
0;​​​​(*Variableplacementا*)​​X=X[ξ_]:=f[ξ];​​Y=Y[ξ_]:=D[f[ξ],{ξ,1}];​​X'[ξ_]:=D[f[ξ],{ξ,1}];​​Y'[ξ_]:=D[f[ξ],{ξ,2}];​​​​(*Definitionofqqandss*)​​qq=(h[X]+g[X]*Y[ξ])*Sum[a[i][X[ξ]]*
i
Y
[ξ],{i,0,m}];​​ss=Sum[D[a[i][X[ξ]],ξ]*
i+1
Y
[ξ],{i,0,m}]+Sum[i*a[i][X[ξ]]*
i-1
Y
[ξ]*(Y'[ξ]),{i,0,m}];​​​​(*Equationofequalityofqqandss*)​​eq2=qqss​​​​(*DeterminethecoefficientsofthepowersofYandsolveform=1*)​​m=1;​​a1[ξ_]=1;​​g[X_]=0;​​h[X_]=A*X+B;​​​​(*Assumingh(X)*)(*Takingtheintegraloftheequationfora0(X)*)​​a0Integral=Integrate[A*X+B+(1-(c/(1-
2
c
)))X,X]+D0;​​​​(*Substitutinga0(X),a1(X),h(X)intothefinalequationی*)​​a0[ξ_]:=1/2*(A*X)+(B+(c/(1-
2
c
)))*X+D0;​​​​(*EquationfordifferentpowersofX*)​​finalEq=a1[X[ξ]](α/(1-
2
c
))*X+β(1-
2
c
)
3
X
h[X]*a0[X];​​​​(*SolvingthesystemofnonlinearequationsforcoefficientsA,B,D0*)​​solutions=Solve[{finalEq},{A,B,D0}];​​
Out[]=
(h[Null]+g[Null]Null[ξ])
m
∑
i=0
i
Null
[ξ]a[i][Null[ξ]]
m
∑
i=0
i
-1+i
Null
[ξ]a[i][Null[ξ]]
′′
f
[ξ]+
m
∑
i=0
1+i
Null
[ξ]
′
a[i]
[Null[ξ]]
′′
f
[ξ]
​
Out[]=
αf[ξ]
-1+
2
c
+
β
3
f[ξ]
-1+
2
c
-
c
′
f
[ξ]
-1+
2
c
+
′′
f
[ξ]0