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.