Ask Uncle Colin: Solve this!
Dear Uncle Colin,
I need to find the largest solution to
and I don’t really know where to start. Any ideas? - Some Options Look Virtually Equal
Hi, SOLVE, and thanks for your message!
That is something we in the trade call ‘not a very nice equation to solve’. It’s quite unusual for mixtures of exponentials and trig functions to have simple roots, and this, I’m afraid, is one of the awkward cases.
Finding bounds
We can put some bounds on it, though: we know that when
Similarly, the largest solution must be larger that
Getting closer
At this point, making an estimate of
Wolfram|Alpha puts it at -0.5885…, and doesn’t suggest any nice exact form for the answer.
Hope that helps!
- Uncle Colin
* Edited 2018-01-12 to clarify what kind of case it was. Thanks, Adam!