Numerically Solving a System of Coupled Integro-Differential Equations

Question

I want to solve a system of coupled Integro-Differential equations in the form below:

With $k_{min} = 10^{-14} , k_{max} = 10^{-7}$ $\eta_{min} = 10^{13} , \eta_{max} = 10^{15}$

And there is already an interesting answer that is very similar to my problem. However, I just don't know how to apply it to my problem as it seems DSolveValue can't handle systems of IDEs at once. As an example, this simple case works:

\[Lambda] = 1;
PHI = DSolveValue[{\[Phi][x] == 
    3 + \[Lambda] Integrate[
       Cos[x - s] \[Phi][s], {s, 0, Pi}]}, {\[Phi][x]}, x]

{(3 (-2 + \[Pi] - 4 Sin[x]))/(-2 + \[Pi])}

But adding a second equation doesn't work:

\[Lambda] = 1;
PHI = DSolveValue[{\[Phi][x] == 
    3 + \[Lambda] Integrate[Cos[x - s] \[Phi][s], {s, 0, Pi}], 
   y[x] == x}, {\[Phi][x], y[x]}, x]

out:

\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Pi]\)]\(\(Cos[
      s - x]\ \[Phi][s]\) \[DifferentialD]s\)\), x}

Can anyone point me in the right direction to solve this system of IDEs?