Solve two equations simultaneously?

Questions
,
1

I have two equations involving sigma notation and want to solve for constants C1 and C2 simultaneously. How can I solve these equations? Any help will be highly appreciated.

Equations.nb (8.83 KB)

2

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.

3

thanks for your explanation. I forgot to mention one thing that Y=1 after differentiating. But I tried to solve it I could not get the answer.
D[Eq1, Y] == 0 at Y == 1

4

That does not change things significantly. Now the condition

sum[\[Phi]2[Y]] sum[
\!\(\*SuperscriptBox["f1", "\[Prime]",
MultilineFunction->None]\)[1]] - sum[\[Phi]1[Y]] sum[
\!\(\*SuperscriptBox["f2", "\[Prime]",
MultilineFunction->None]\)[1]] == 0

must be fulfilled.

Perhaps you could check it numerically with some values for your parameters.

Equations _OP_4_CP_.nb (61.8 KB)