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I have problem of eliminating some variables from a set of equations, which contain logarithmic function, in order to obtain a differential equation.

These are the parameter and functions in eq1:

Ξ = (EE*((1 + B KK) Sqrt[M] Log[(1 - x0)/(1 - xw)] - 
B Δ Subscript[Θ, ∞])^2)/(\
Δ^2*(Γ*(1 - xw)*(Log[(1 - x0)/(1 - xw)] - 
Δ/Sqrt[M]*B*Subscript[Θ, ∞]) - 
KK*B*Subscript[Θ, ∞]));

g1 = ((Δ*Γ*(xi - xw) + KK*Sqrt[M]*
Log[(1 - x0)/(1 - xi)])*(Subscript[Θ, ∞] -\
Γ*(xi -xw)))/((Subscript[Θ, ∞] - \
Γ*(xi - xw))*Δ*B - 
Sqrt[M]*(1 + KK*B)*Log[(1 - x0)/(1 - xi)]);

g2 = ((Δ*Γ*(xi - xw) + 
KK*Sqrt[M]*Log[(1 - x0)/(1 - xi)])*(1 + KK*B))/(Δ*
B*(Subscript[Θ, ∞] - 
Γ*(xi - xw)) - (1 + KK*B)*Sqrt[M]*Log[(1 - x0)/(1 - xi)]);

These are the equations to be combined:

eq1 = H'[T] == 
1/Ξ*(EE*(Γ*(xi - xw) - B*g1))/(KK + g2);

eq2 = H[T] == (Δ*Γ*(xi - xw) + 
KK*Sqrt[M]*Log[(1 - x0)/(1 - xi)])/(Δ*
B*(Subscript[Θ, ∞] - 
Γ*(xi - xw)) - Sqrt[M]*(1 + KK*B)*Log[(1 - x0)/(1 - xi)]);

eq3 = Xp == (Log[(1 - x0)/(1 - xw)] (-Sqrt[M] (KK (1 + B KK) Sqrt[M] + 
(-1 + xw) Γ Δ) Log[(1 - x0)/(1 - xw)] + 
B (-1 + xw) Γ Δ^2 Subscript[\
Θ, ∞]))/((1 + B KK) Sqrt[M]*
Log[(1 - x0)/(1 - xw)] - 
B Δ Subscript[Θ, ∞])^2;

Eliminate[{eq1, eq2, eq3}, {EE, Δ, Γ, M, 
KK, B, Subscript[Θ, ∞], xi, xw, x0}]

The long running time implies that there should be some tricks to deal with suck kind of problem. I think the issue is lying in the logarithmic function. This is a very similar answered question.

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  • Solving one by one might help. Solve the first equation for any parameter (say \xi), substitute that parameter in equation system, then Simplify, and try again. This is helping Mathematica to eliminate. Since Mathematica tries to solve this equation system at once, it might be slow. – gurluk Dec 03 '15 at 08:02
  • Why have you asked again? http://mathematica.stackexchange.com/q/101004/5478 – Kuba Dec 03 '15 at 08:08
  • Hi, @Kuba, I am so sorry. I feel the previous post is too long. I just want to change it into a more concise form. Do you suggest me to delete it :) Could you pls give me some hints to the problem? Many thanks! – Enter Dec 03 '15 at 12:04
  • @gurluk, Let me have a try and then give a feedback! – Enter Dec 03 '15 at 12:06
  • Hi,@gurluk, I follow your advice to do these things: First not assign \[CapitalXi], Solve[eq1, xi] // FullSimplify to obtain xi, then plug it in eq2, (eq2 /. xi ->1 + E^((\[CapitalDelta] \[CapitalXi] Derivative[1][H][T])/(EE Sqrt[M])) (-1 + x0)) // FullSimplify, and assign the resulting H[T] as eq4 = H[T] (note the expression has not be pasted here), the process is fine up to now, then I try Eliminate[{eq3, eq4}, x0], however, it is slow again...help – Enter Dec 04 '15 at 03:08

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