this is my first post. I am trying to figure out this "exploration" for my high level college math class and I think trying to code in excel is messing me up.
"1. Consider the “simple” nonautonomous differential equation dy/dt = e^t*sin(y). Sketch the graph of the solution to this equation that satisfies the initial condition y(0) = 0.3. 2. Use Euler’s method with a step size 1t = 0.3 to approximate the value of the previous solution at y(10). It is probably easiest to use a spreadsheet to carry out this method. How does your numerical solution compare to the actual solution? 3. Repeat the previous calculation with step sizes 1t = 0.001,0.002, and 0.003. What happens now? This behavior is called sensitive dependence on initial conditions, the hallmark of the phenomenon known as chaos. 4. Repeat step 2 but now for initial conditions y(0) = 0.301 and y(0) = 0.302. Why is this behavior called senstitive dependence on initial conditions?"
So I've solved the equation to be y(t)= 2(arccot(17.9858e^(-e^t)) and I've put =2(ACOT(17.9858*EXP(-EXP(t)))) into excel for different values of t and different initial conditions but everything gives my y(10)=3.141593, which I don't think should be happening right? the different initial conditions should be sending it into chaos.
Yes, the exact solution converges towards the next stable fixed point which is at $y=\pi$.
I'm not sure I would call the observations about the numerical solutions "chaos". You run into stability issues. For the Euler method stability requires that $L\Delta t\le 2$ at the most, better smaller like $\le0.5$, where $L$ is a $y$-Lipschitz constant of the equation. Here that gives $L=e^t$, which depends on $t$ and becomes rather large at $t=10$, $e^{10}=22026.5$
One can of course consider the Euler process at a fixed step size as a discrete dynamical system that has its own kind of chaos. But that has nothing to do with the properties of the exact solution