An inverse problem; does an analytic solution exist?

[Vector_Frank]

x = a*sin(b*t + c) + d*t

a, b, c, and d are arbitrary constants. Is there an analytic solution for t for a given x?


In the graph: a = 1, b = 1, c = 1, and d = 0.5.

This inquiry is similar to another recent one which I have read:

Inverse formula of a difference of cosines

Hi, my problem is the following: I need to find the inverse formula of this equation to get the "t" inside of the cosine. So, to be as clear as possible, in the end i should have the equation: "t = ...". x = cos(t) - cos(15t) Thanks to anyone will try to help me.

www.freemathhelp.com

Responses there haven't answered my question.

Background: I am programming a molecular dynamics simulation for diatomic gasses. A molecule may
be translating and rotating. I need to determine the time of its impact with a boundary.

I realize that the problem may have multiple solutions. The earliest t is the solution that I need.
I suspect that no analytic solution exists and that I will need to use numerical methods.
Before I begin programming, I want to verify that I am not missing a simpler solution.

 

[Deleted member 4993]

x = a*sin(b*t + c) + d*t

a, b, c, and d are arbitrary constants. Is there an analytic solution for t for a given x?

View attachment 16903
In the graph: a = 1, b = 1, c = 1, and d = 0.5.

This inquiry is similar to another recent one which I have read:

Inverse formula of a difference of cosines

Hi, my problem is the following: I need to find the inverse formula of this equation to get the "t" inside of the cosine. So, to be as clear as possible, in the end i should have the equation: "t = ...". x = cos(t) - cos(15t) Thanks to anyone will try to help me.

www.freemathhelp.com

Responses there haven't answered my question.

Background: I am programming a molecular dynamics simulation for diatomic gasses. A molecule may
be translating and rotating. I need to determine the time of its impact with a boundary.

I realize that the problem may have multiple solutions. The earliest t is the solution that I need.
I suspect that no analytic solution exists and that I will need to use numerical methods.
Before I begin programming, I want to verify that I am not missing a simpler solution.

Try transforming the given equation first by rotating the axes so that you get:

u = E * sin(F*v + G) ............... eliminating the linear term.

 

[Dr.Peterson]

When the variable exists both inside and outside of a transcendental function (the sine here), my assumption is that it can't be solved analytically, unless there is something very special about it.

The fact that there can be any number of solutions (even hundreds or thousands) complicates your problem further; that was the main issue in the thread you referred to, but is secondary in your problem.

Subhotosh's suggestion is interesting, but I can't picture it working, in part because I can't see it yielding a variable but finite number of solutions. In particular, rotation will not turn the curve into a pure sinusoid.

 

[Vector_Frank]

Thank both of you for your replies.

Subhotosh Kahn, unfortunately rotating axes will not yield an expression is the form you showed. But, it's an interesting approach.

Dr. Peterson, your response confirmed my suspicion. I must resort to numerical methods.

Thanks again to each of you.