1. I'm a high school student, and I am working on a physics project.
But I've got a calculus problem when conducting some ananlysis and trying to derive a formula.

The following are the equations of my derivation.

I want to find out x(t) and y(t).
(a, b, c, f are the parameters of my physics model.)
I've asked my math teacher and she said this problem can be solved but it's very complex.

So please help me get x(t) and y(t) or just tell me how to solve it. (What math tools should I use?)

Thank you!

2.

3. The fact that your solution for x(t) and y(t) depends on x(t), y(t) and their derivatives means that you have a non-linear problem. This makes it highly unlikely that there's an analytic solution, which means a solution would have to be obtained numerically. If it's an analytic solution that you desire however then a good place to start is to go back over your derivation and making some simplifying assumptions and exploit any symmetry in the problem.

May i ask what kind of a system this is a model of?

4. Wow! You are just 15 (or 16), but you know really much.
I have few concepts about calculus so I'm not sure if the result I have derived is reasonable or not. Sorry.

Yes. it's a non-linear problem. I want to research on the shape of a phenomenon about liquid. So I put the shpae onto the coordinate system with x-axis and y-axis, Then, I derived respectively the position on the x-axis and y-axis and use the function of t (time) to express it. The shape is a curve so I need cosɵ and sinɵ.

cosɵ= x'(t)/[(x'(t)^2+y'(t)^2)^1/2]
sinɵ=y'(t)/[(x'(t)^2+y'(t)^2)^1/2]

That's why I got x'(t) and y'(t) in my equations. ......
(I will PM you the details if you are available. Thank you!)

I wonder whether you mean I need to have some numerical analysis to slolve my problem.
Need I use some experimental data to derive it?

5. Originally Posted by kevinchang
Yes. it's a non-linear problem. I want to research on the shape of a phenomenon about liquid. So I put the shpae onto the coordinate system with x-axis and y-axis, Then, I derived respectively the position on the x-axis and y-axis and use the function of t (time) to express it. The shape is a curve so I need cosɵ and sinɵ.

cosɵ= x'(t)/[(x'(t)^2+y'(t)^2)^1/2]
sinɵ=y'(t)/[(x'(t)^2+y'(t)^2)^1/2]

That's why I got x'(t) and y'(t) in my equations. ......
(I will PM you the details if you are available. Thank you!)
I'm still not quite sure what it is that you're modelling, but that's not really what's important. It seems to me that your expressions for sine and cosine define them to be the rate of change of the x-coordinate (or y-coordinate) with respect to the arc-length of the curve, maybe with context makes this necessary. Otherwise a much simpler approach to the problem would be to stick with,.

Originally Posted by kevinchang
I wonder whether you mean I need to have some numerical analysis to slolve my problem.
Need I use some experimental data to derive it?
Numerical Analysis would be the way to go, however what you have is a non-linear integral equation. A quick Google search has convinced me that, numerical solutions to non-linear integral equations are not a simple thing to find.

In short your maths teacher was right on the money in saying that any solution to these equations would be very complex. However i'd go back over the derivation of the equations and double check your working/ re-examine the reasoning.

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