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Can anyone please help me fined (if it is possible) a closed-form solution or an approximation for the solution for the following equation (x is the only variable):

$$\frac{((a-1)b^{x+2}-(b-1)a^{x+2}+b-a)v+((a-1)b^{x+2}-(b-1)a^{x+2})u}{b-a}=0$$

What is known is:

  1. a solution exists (the problem is finding a closed-form expiration).

  2. $x\ge0$.

  3. $a,b,v,u$ are parameters such that $0<a<b<1$, $v>0$, $u>0$.

Even an approximation for the solution will help. Since an expression is needed then numerical methods are not helpful here.

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You can immediately solve for $y=(a-1)b^{x+2}-(b-1)a^{x+2}$ which occurs linearly twice in this expression, in the form:

$$y=-\frac{v(b-a)}{u+v}$$

However, you cannot get $x$ from $y$ in closed form unless for special values of $a$ and $b$, because the equation above has no solution in terms of elementary functions. Consider $b^x=u$ and $a^x=u^{p}$ where $p=\ln a/\ln b>1$. Then, you have a "polynomial" with a real exponent:

$$y=(a-1)b^2 u - (b-1)a^2 u^{p}$$

For numerical solution, I would start here (it's a reasonable expression where you can at least estimate the number and nature of solutions). Since you are asking for an analytical solution, this won't help much, because $p$ can be in principle anything from $1$ to $\infty$. Unless you have any other hints of the values -- if $p$ is very big, you can probably ignore the first term and get an analytical approximation. If $p$ is very close to $1$ ($a$ and $b$ very close together), you can do series expansion of all terms.

For the last case, you would take the expression for $y$ and compute the derivative with respect to $b$, and then take $b=a$, to get something like this:

$$y\approx y|_{b=a} + \frac{\partial y}{\partial b}|_{b=a} (b-a)$$

where of course you would have the painful problem of differentiating the $p$ term in the exponent. Unfortunately, I tried and even this equation contains combination $u\ln u$ and is therefore not solvable in terms of standard functions (you need Lambert's W function).

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  • $\begingroup$ Here is a link that tells about explicit solution of transcendental equations: "hindawi.com/journals/ijem/2015/523043" $\endgroup$ – Awe Kumar Jha Nov 27 '18 at 15:37
  • $\begingroup$ How do you apply it? $\endgroup$ – Y.L Nov 27 '18 at 15:48
  • $\begingroup$ Thanks @orion, regarding your last sentence, if $a$ and $b$ are very close together, can you please show how it can be done? $\endgroup$ – Y.L Nov 27 '18 at 15:58
  • $\begingroup$ That link shows a numerical method that requires the function to be evaluated in a lot of points before "fitting" a polynomial to it. There is no miracles or free lunch. $\endgroup$ – orion Nov 27 '18 at 16:02
  • $\begingroup$ Thank you @orion but I get stuck again at your last sentence lol. How do you apply this Lambert's W function method here? :) $\endgroup$ – Y.L Nov 27 '18 at 16:25

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