# Physics 212, 2018: Lectures 8

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 ${\displaystyle dt}$.
You will write a different differential equation and will solve it, 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.