# Physics 212, 2019: Lecture 9

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## Runge-Kutta 2 integration

How do we solve systems on nonlinear ODEs in the first place? The Euler method, which we introduced before, is good. But how accurate is it? We will talk about ${\displaystyle O(dt^{n})}$ notation, and we will show that the Euler method is ${\displaystyle O(dt)}$. We will verify this by using the scripts that I have uploaded below. We will run the simple Euler method of solving differential equations on the malthusian grows problem with different ${\displaystyle dt}$, and show by plotting the error at the end that the error goes down in proportion to ${\displaystyle dt}$.

We then will talk about Runge-Kutta methods of solving differential equations. I will add additional notes later on, but for now please use these lecture notes http://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node5.html . Additionally, we discussed how RK2 method is a prediction-correction method.

Use a different differential equation from those provided to show that Runge-Kutta 2 has error ${\displaystyle O(dt^{2})}$, and the scaling of the error is not a scaling due to the equation being solved, but largely due to the methods used to solve it. Do this in a Jupyter notebook.