[?] Solve equation t + Sin[w t] = a in t?

Questions
1

Hi,

Is there a way to solve (also in an approximate way) an equation of this type with Mathematica?

t + Sin[w t] = a in variable t.

Using Solve[t + Sin[w t] == a, t] can’t get the solution.

Thanks!

2

General analytical solution for transcendental equation is impossible.

Approximation by numerics:

     w = 1; a = 1;

     sol = Solve[t + Sin[w t] == a, t, Reals];
     sol // N;
     (*{{t -> 0.510973}} *)

     sol2 = NSolve[t + Sin[w t] == a && -10 < t < 10, t, Reals]
     (*{{t -> 0.510973}} *)

     sol3 = FindRoot[t + Sin[w t] == a, {t, 1}]
     (*{t -> 0.510973}*)

Regards M.I.

3

Thanks!

4

Closely related to Kepler’s Equation.

5

DynamicModule[{k}, Panel[
Column[{
Row[{“a”, " ", Slider[Dynamic[k], {1, 9, 0.3}]}],
With[{a = k},
Dynamic@ContourPlot[t + Sin[w t] == a, {t, 0, 10}, {w, -5, 5},
PlotPoints → 25, ImageSize → 300,
PlotLabel → “w vs.t given a”, FrameLabel → {“t”, “w”}
]]}]
]
]

import.png
import.png460×521 7.43 KB

6

Interesting:)

7

Very nice, didn’t know this feature!