# algebraic manipulation – Where I am wrong?

I have an ODE:
(Eq.4)

STEP 1: Assume that the solution of ODE can be expressed by

``````ClearAll("Global`*")
m = 2;
U((Xi)_) =
Sum(Subscript(l, i) Exp(-(CapitalPhi)((Xi))^i), {i, 0, m})
auxEQ = Exp(-(CapitalPhi)((Xi))) + (Mu) Exp((CapitalPhi)((Xi)))
+ (Lambda)
``````

STEP 2: By substituting Eq. (8) into Eq. (4) and using the auxiliary equation in Eq. (9), and then collecting all terms with the same order of $$exp(−phi(xi))$$ together, the left hand
side of Eq. (4) is converted into a new polynomial in $$exp(−phi(xi))$$. Setting each coefficient of this polynomial to
zero, yields a system of algebraic equations for $$l_0,l_1,ldots l_m, lambda$$ and $$mu$$.

`````` ODE = 3 U((Xi)) D(U((Xi)), {(Xi), 2}) -
3 (D(U((Xi)), (Xi)) )^2 + U((Xi))^3
newODE = ODE //. {D((CapitalPhi)((Xi)), (Xi)) -> auxEQ,
D((CapitalPhi)((Xi)), {(Xi), 2}) -> D(auxEQ, (Xi))};
algebraicSYSTEM=CoefficientList(newODE,
Table(E^-n (CapitalPhi)((Xi)), {n, 0, m})) == 0 // LogicalExpand
``````

I should get the following algebraic system:

But my Mathematica code gives a different result.