How to solve this equation for x ??
A sin (x − B)=exp Cx
Where A,B, C ...are constants
You have asked: How to solve this equation for x ??
Since no-one else has attempted a reply, I will.
What follows is a 'starter', which hopefully you will be able to complete.
Equations like this will have an infinity of solutions, usually expressed as a power series.
My attempt is based on that idea . . .
Consider the trig. identity:
a.sinx - b.cosx = √(aČ + bČ).sin(x - α), where tanα = b/a and is 1st. quadrant.
For this Q., let √(aČ + bČ) = A; α = B.
Then we now have:
a.sinx - b.cosx - e^(cx).
Using the power series for sinx, we obtain:
a.sinx = a[x - x^2/2! + x^4/4! + . . . .]
and similarly for cosx:
b.cosx = b[x - x^3/3! + x^5/5! - . . . .]
and for e^cx:
e^cx = 1 + cx + (cx)^2/2! + (cx)^3/3! + . . . .
Hence, we finally arrive at:
a[x - x^2/2! + x^4/4! + . . . .] - b[x - x^3/3! + x^5/5! - . . . .] = 1 + cx + (cx)^2/2! + (cx)^3/3! + . . . .
I'll leave the rest up to you (I did say that what follows is a 'starter' ).
Hint: next: move the RHS to the LHS (changing signs accordingly) and simplify the result.
An infinite power series will be produced.
Hopefully, you can take things from there . . . .
Hope that helps!
Al. / Skywave
June 18, 2017
This individual posted more than once with variations on the theme of this particular post ...
Find out solution
... which is probably why no one bothered to respond to this one.
Skeeter: "This individual posted more than once with variations on the theme of this particular post."
I didn't know that. Moreover, did I not read somewhere that doing suchlike (posting same Q. more than once) is against a Forum rule?
Al.
Forum Rules
rule #10
Because we only have one moderator, topsquark ... the site administrator has not authorized additional moderators for whatever reason(s) since he took over this site years ago. Suffice to say, it is what it is & we deal with it.