That seems to be difficult. Your system can be written as
Eq1 = C[2] sum[f1[Y]] + C[1] sum[f2[Y]]
Eq2 = C[2] sum[\[Phi]1[Y]] + C[1] sum[\[Phi]2[Y]]
where I used obvious abbreviations.
Now your 1st condition translates to (assuming the summation and differentiation may be interchanged)
dEq1 = Eq1 /. {f1[Y] -> D[f1[Y], Y], f2[Y] -> D[f2[Y], Y]}
with your 2nd condition you arrive at
In[14]:= Solve[{dEq1 == 0, Eq2 == 0}, {C[1], C[2]}]
Out[14]= {{C[1] -> 0, C[2] -> 0}}
So to get nontrivial solutions the following must be true
In[16]:= Det[( {
{sum[f1'[Y]], sum[
\!\(\*SuperscriptBox["f2", "\[Prime]",
MultilineFunction->None]\)[Y]]},
{sum[\[Phi]1[Y]], sum[\[Phi]2[Y]]}
} )] == 0
Out[16]= sum[\[Phi]2[Y]] sum[
\!\(\*SuperscriptBox["f1", "\[Prime]",
MultilineFunction->None]\)[Y]] - sum[\[Phi]1[Y]] sum[
\!\(\*SuperscriptBox["f2", "\[Prime]",
MultilineFunction->None]\)[Y]] == 0
But I am afraid this is not easy to check.