Eliminate functions from system of ordinary differential equations

Question

I have the following system of ODE

eq1 = -s^2 ep[x] - cp''[x] + ep''[x] - θp''[x];
eq2 = A cp[x] - B cp''[x] + ep''[x] + C θp''[x];
eq3 = E cp[x] + F cp[x] + s ϵ ep[x] + G ep[x] + s θp[x] + K θp[x] - θp''[x];

A, B, C, E, F, G and K are constants. I need to eliminate both ep[x] and cp[x] from the equations so I solved eq1 and eq2 and got ep[x] and cp[x] by:

 Solve[eq1 == 0, cp[x]] // Simplify
 Solve[eq2 == 0, cp[x]] // Simplify  
 eq3 // Simplify

Then how I can eliminate cp[x] and ep[x] and their derivatives from eq3? Using the substituting /. rule will remove cp[x] and ep[x] but not their derivatives. I should obtain a 6th order differential equation of θp only.

Answer 1

This question is actually very similar to this one. It can be solved in the following way:

Eliminate[{eq1 == 0, D[eq1, x, x] == 0, eq2 == 0, D[eq2, x, x] == 0, eq3 == 0, 
   D[eq3, x, x] == 0, D[eq3, {x, 4}] == 0}, {cp[x], ep[x], cp''[x], ep''[x], cp''''[x], 
   ep''''[x]}] // Simplify

(*  (s^2 (C (E + F) + B (K + s)) + A (G + K + s (1 + s + ϵ))) 
       θp''[x] + (-1 + B) θp''''''[x] == 
   A s^2 (K + s) θp[x] + (A + E + C E + F + C F + B G + C G - K + B K - 
          s + B s + B s^2 + B s ϵ + C s ϵ) θp''''[x]  *)

We can also make use of the hidden syntax of Solve. (In many cases it performs better than Eliminate):

Equal @@@ First@
   Solve[{eq1 == 0, D[eq1, x, x] == 0, eq2 == 0, D[eq2, x, x] == 0, eq3 == 0, 
     D[eq3, x, x] == 0, D[eq3, {x, 4}] == 0}, θp[x], {cp[x], ep[x], cp''[x], 
     ep''[x], cp''''[x], ep''''[x]}] // Simplify