Solving time dependent differential equation and plotting

pleasehelpmeno

Hi i have the differential equation

[itex]\frac{d^{2}}{dt^{2}}X(t) +(A+B\frac{sin^{2}(mt)}{mt})X(t)[/itex] I have tried by hnd to solve this and am getting knowhere does anyone know how to solve it and then plot X against t (where the constants A, B and m will be arbitrarily added), possibly using maple? I know that it is related to the mathieua equations and have found this.

Thanks

HallsofIvy

Do you need an "analytic" solution? If you just want to graph the solution, I would recommend a "Runge-Kutta" numerical solution.

pleasehelpmeno

I just want to get a X= f(t) equation that i can then plot, for t values

MathematicalPhysicist

I guess you mean:

[tex]\frac{d^{2}}{dt^{2}}X(t) +(A+B\frac{sin^{2}(mt)}{mt})X(t)=0[/tex]

In which case as always multiply by dX/dt

[tex] 1/2 \frac{d}{dt}(\frac{dX(t)}{dt})^2 + (A+B\frac{sin^{2}(mt)}{mt})/2 \frac{d}{dt}(X^2(t)) = 0[/tex]

Now there's a problem, you need to integrate by parts to get the next equation:

[tex] 1/2 (\frac{dX(t)}{dt})^2 + (A+B\frac{sin^{2}(mt)}{mt})/2 X^2(t) - \int X^2(t)/2 (B\frac{d}{dt}( \frac{sin^{2}(mt)}{mt}) dt = E[/tex]

So now we get a mixed differentail-integral equation, which I am not sure how to approxiamte.

JJacquelin

You can't on the way you are thinking of.
The solutions cannot be expressed as a combination of a finite number of usual functions. In similar cases, some special functions were defined in order to represent the solutions on a closed form (see the example below, in case of a simpler equation). As far as I know, it was not done in the case of your equation. So, do not expect to find an exact analytical equation X=f(t).
Of course, this ODE can be solved and the result can be plot, thanks to numerical computation methods.
Another way for Physicians is to replace the equation by another simpler one which can be analyticaly solved, if the physical phenomena can be modeled with some approximations on some limited ranges of the parameters.

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