# Solving time dependent differential equation and plotting

In summary, the conversation discusses an unsolvable differential equation and suggests using numerical methods or approximations to find a solution. It also mentions that the equation is related to the mathieu equations and cannot be expressed as a combination of usual functions. The conversation ends with a suggestion to use numerical computation methods or approximations to find a solution.
Hi i have the differential equation

$\frac{d^{2}}{dt^{2}}X(t) +(A+B\frac{sin^{2}(mt)}{mt})X(t)$ 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

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

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

I guess you mean:

$$\frac{d^{2}}{dt^{2}}X(t) +(A+B\frac{sin^{2}(mt)}{mt})X(t)=0$$

In which case as always multiply by dX/dt

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

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

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

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

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

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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## 1. What is a time dependent differential equation?

A time dependent differential equation is a mathematical equation that describes the relationship between a function and its derivatives with respect to time. It is used to model dynamic systems where the rate of change of a variable depends on time.

## 2. How do you solve a time dependent differential equation?

The process of solving a time dependent differential equation involves finding a function that satisfies the equation. This can be done through various methods such as separation of variables, substitution, or using numerical techniques like Euler's method. The solution can then be plotted to visualize the behavior of the system over time.

## 3. What is the importance of solving and plotting time dependent differential equations?

Solving and plotting time dependent differential equations allows for the prediction and understanding of the behavior of dynamic systems. This is crucial in many scientific fields such as physics, engineering, and economics, where the behavior of systems over time is of great interest.

## 4. Can time dependent differential equations be solved analytically?

Some time dependent differential equations can be solved analytically, meaning a closed-form solution can be obtained. However, many equations do not have analytical solutions and require numerical methods to approximate the solution.

## 5. How do you plot the solution of a time dependent differential equation?

To plot the solution of a time dependent differential equation, the dependent variable is typically plotted on the y-axis and the independent variable (time) on the x-axis. The solution can be plotted using software such as MATLAB or Python, or manually by creating a table of values and plotting them on a graph.

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