# Unable to solve equation for a variable by any method

I am looking to solve for the value of r when the derivative of the expression

 (1/ r^2) (1 - ((2*GM)/(c^2*r)) Integrate[t*Exp[(-t)], {t, 0, r/Sqrt[β]}])


equals zero.

But I have been unable to use the Solve, FindRoot, or Reduce to find a value for r. Mathmatica keeps returning

Solve::nsmet: This system cannot be solved with the methods available to Solve.

G, β, M, and c are all constants.

Please, any help would be much apreciated.

• Transcendental equations of this sort generally do not have closed form solutions. As I said in your other question, you need a starting guess for FindRoot[]. For Solve[]/Reduce[], unless you have specific values for your constants and a region of interest, they can't do much. – J. M. will be back soon Feb 21 '16 at 13:28
• I assumed [Beta] was meant to be \[Beta] (i.e. β). If I'm wrong, you can roll back the edit or fix it yourself. FYI: You can format inline code and code blocks by selecting the code and clicking the {} button above the edit window. The edit window help button ? is also useful for learning how to format your questions and answers. You may also find this this meta Q&A helpful – Michael E2 Feb 21 '16 at 14:00
• GM is not the same as G*M – m_goldberg Feb 21 '16 at 17:11

## 1 Answer

It would be better when writing this equation if the values of the constants GM, c and β were supplied.

I will set them all to the value one but it should work for any real coefficients.

First the integral is evaluated. This is generally a good practice so that in downstream processing the integral is not repeatedly evaluated.

Integrate[t*Exp[(-t)], {t, 0, r/Sqrt[β]}]

(* 1 - E^(-(r/Sqrt[β])) (1 + r/Sqrt[β]) *)


Next place that result into your original equation

eq1 = (1/r^2) (1 - ((2*GM)/(c^2*r)) 1 -
E^(-(r/Sqrt[β])) (1 + r/Sqrt[β]))

(* (1 - (2 GM)/(c^2 r) -
E^(-(r/Sqrt[β])) (1 + r/Sqrt[β]))/r^2 *)


and take the derivative

eq2 = D[expr, r];


yielding

-((2 (1 - (2 GM)/(c^2 r) -
E^(-(r/Sqrt[β])) (1 + r/Sqrt[β])))/r^3) + ((2 GM)/(
c^2 r^2) - E^(-(r/Sqrt[β]))/Sqrt[β] + (
E^(-(r/Sqrt[β])) (1 + r/Sqrt[β]))/Sqrt[β])/r^2


Next substitute numerical values

eq3 = eq2 /. {GM -> 1, c -> 1, β -> 1}

(* -((2 (1 - 2/r - E^-r (1 + r)))/r^3) + (-E^-r + 2/r^2 +
E^-r (1 + r))/r^2 *)


Finally (the most important step), apply J.M.'s suggestion of using FindRoot to determine the value.

FindRoot[eq3, {r, 4}]

(* {r -> 3.9647} *)

• Jack, it is likely that the poster meant GM to stand for G*M, since GM itself is not in the list of constants mentioned in the OP. – MarcoB Feb 21 '16 at 19:06
• Thank you all very much for your help. – Michael Lerner Feb 22 '16 at 2:23
• IT WORKED OUT. Thanks to all of your help I was able to input the values for the constants and acheive the results I needed. I cannot thank you all enough, this was for my seinor undergraduate thesis for physics and you all helped me to finish it on time. – Michael Lerner Feb 22 '16 at 7:32