Thread: help with a non-linear ode with unusual properties that might aid in the solution

Hello, I'm a first time poster and a hobbyist who is trying to model a system,
I'm attempting to solve this ODE:
y' = cos(cx)sin(yx) where y is a function of x and c is a constant

I can't seem to get Mathematica (I just got a trial version) to do much with it and solving it on it's own is pretty hard.

Something that may be helpful: I have also considered that since, by the chain rule, if we make
g(x) = yx then
g'(x) = cos(cx). this is the little trick that might make this work


Using the product rule g'(x) =y'x + x'y = y'x+y = cos(cx) which can be rewritten as:
x*cos(cx)sin(yx)+y=cos(cx)

This is no longer an ODE. It's well, just an equation. But I'm nut sure how to simplify it into something i can graph, even though i've tried (using Mathematica) graphing it using the above and the complex exponential equivalent of the equation.

Help?

Originally Posted by chadmcgrath

Hello, I'm a first time poster and a hobbyist who is trying to model a system,
I'm attempting to solve this ODE:
y' = cos(cx)sin(yx) where y is a function of x and c is a constant

I can't seem to get Mathematica (I just got a trial version) to do much with it and solving it on it's own is pretty hard.

Something that may be helpful: I have also considered that since, by the chain rule, if we make
g(x) = yx then The right hand side contains both on x and y. So writing g(x) as if g depends only on x is incorrect.
g'(x) = cos(cx). this is the little trick that might make this work This is incorrect again.


Using the product rule g'(x) =y'x + x'y = y'x+y = cos(cx) which can be rewritten as:
x*cos(cx)sin(yx)+y=cos(cx)

This is no longer an ODE. It's well, just an equation. But I'm nut sure how to simplify it into something i can graph, even though i've tried (using Mathematica) graphing it using the above and the complex exponential equivalent of the equation.

Help?

Hi chadmcgrath,

I tried your differential equation with both MATLAB and Wolfram Alpha, but both failed to give a solution. However it could be easily seen that, y=0 is solution.

Can you use the small angle approximation? That is, that if the product yx is sufficiently small, then sin(yx) is approximately yx?

Thanks for replying, I wasn't exactly clear on how making g[x] = yx is not allowed. In the original post, I say that y is a function of x (however it might not have been entirely apparent, and it might not help that I was mixing notations up now that I look back at it.). Maybe I'm the wrong track with that in any event but that has often worked for me in the past.
In that case I figured the derivative of sin(g[x]) is g'[x]Sin(g[x)). In this case yielding g'[x] = cos[x] as a possible solution.

This works(I think) for instance, in the case where the the left hand side is not equal to y'. Instead f'[x] = cos[x]Sin[yx] you can plug in y=sin[x]/x as a solution.

Is that all wrong?

Thanks Ackbeet, Good idea, but in this case both variables could get very large.

The substitution $\displaystyle g=yx$ leads to $\displaystyle g'=x y'+y,$ and hence

$\displaystyle x g'=x^{2}y'+yx,$ or

$\displaystyle x g'=x^{2}\cos(cx)\,\sin(g)+g.$

Still nasty. I've got Mathematica chewing on it right now. I'm guessing it's probably not going to work. Have you tried numerical solutions?

In what context did this DE show up?

I just wrote a very long reply but lost it. I apologize but I don't think the equation I gave you was smooth. So I'll have to work on the model before submitting something like this again.