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I build a model for our problem, but i cannot get a result from my model. Could anyone give me some idea to solve this formula: $aX+bX^2=e^{cX}$

Thx in advance!

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What is $a$, $b$ and $c$? I assume those are constants and you want to solve for $X$? It seems unlikely that you can get closed formulas for $X$. other than in very special cases. For fixed $a$, $b$ and $c$ any standard numerical method for solving equations should suffice. (Newton's method, bisection, or some variant of those.) – mrf May 22 '12 at 9:54
Newton's method will converge for "most" values of $a,b,c$ but studying precisely each case could be time-consuming. You may start with $(a,b,c)=(1,1,1)$, the function is convex near its negative $0$, and use a homotopy method to reach all other values of $(a,b,c)$. Matlab or Mathematica will solve this for you efficiently too. I do not think there is an explicit solution in terms of roots exponentials and logarithms but I would be greatly interested if someone could point to possible proofs of this, perhaps using methods of differential algebra. – plm May 22 '12 at 10:09

I am by far no expert on this and maybe you should wait for somebody with more expertise, but I can tell you that this equation has no analytical solution and has to be solved numerically (I guess you already know that, given your tag). One of the easiest options is Newton's method. To find a zero of $$f(X)=aX+bX^2-e^{cX}$$ you start with a $X_0$ which should be close to the expected zero and then compute inductively $$X_{i+1}=X_i-\frac{f(X_i)}{f'(X_i)}.$$ The sequence should then converge to a solution. There are multifarious ways to improve this and there are situations where this approach doesn't work, but this is a start.

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