Mathematica Problems solving this differential equation for a Pendulum with Mathematica

AI Thread Summary
The discussion revolves around troubleshooting a Mathematica code for solving a differential equation with boundary conditions. The user initially encountered issues with the code, particularly regarding incorrect initial conditions for the function phi, which led to error messages. After receiving guidance, the user adjusted the initial conditions to include the first derivative of phi. They successfully plotted the solutions for both the exact and approximated differential equations but sought advice on how to differentiate the functions in the plot. A suggested solution included using the PlotLegends option to label the functions clearly. The conversation highlights common challenges faced by beginners in Mathematica and the collaborative effort to resolve coding issues.
Lambda96
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Mathematica does not solve the differential equations
Hi,

unfortunately, I have problems that Mathematica does not solve the differential equation. The task is as follows and it is about the task c

Bildschirmfoto 2023-06-28 um 18.54.02.png


In the Mathematica Notebook, the following was written for task c

"You can use the following two lines of code to produce the solutions of the exact and approximated differential equations with boundary conditions. After that, write a plot[ ] function to sketch the two together."

exact[t_] := Evaluate[phi[t] /. NDSolve[..., phi[t], {t, 0, 20}]];
approx[t_] := Evaluate[phi[t] /. DSolve[..., phi[t], t]];I then tried to calculate the differential equation for equation (4) using the two codes and got the following

Bildschirmfoto 2023-06-28 um 18.58.49.png

I then tried solving the differential equation using only NDSolve and get the following.

Bildschirmfoto 2023-06-28 um 19.04.34.png

I am a beginner in Mathematica, so I do not know what I have done wrong in the two formulas?
 
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As the error messages say, your initial conditions are wrong. You have
Code:
phi[0]==0
and
Code:
phi[0]==Pi/2
which is not meaningful. You should have the first derivative of phi for one of those.
Code:
phi'[0]==...
 
Thanks Dale for your help and for looking over my code 👍

I have now tried plotting the differential equations again, but this time with phi'[0]=0 and with ##\gamma=0.1## and got the following.

Bildschirmfoto 2023-06-28 um 21.44.11.png

Is there any way that I can show in the plot which function is which?
 
Lambda96 said:
Is there any way that I can show in the plot which function is which?
There are different options. Try for instance
Code:
Plot[{exact[t],approx[t]},{t,0,20},PlotLegends->"Expressions"]
 
Thanks DrClaude for your help, this is exactly what I was looking for 👍
 

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