In this report we will be looking at the Heun method. Similar to the Euler method, it will help us approximate differential equations. Unlike Euler's method, this is used to solve second order differential equations. in the simplest terms how the Heun method works, is it takes two estimations, and then takes the average of those two results, to get a better approximation. The Solution i chose was the Chickens and the foxes problem. It relatively simple and easy to follow.so first things first here is the formula we will be using for the chickens and the foxes. $$\frac{dc}{dt} = + .1c - .15fc $$

$$\frac{df}{dt} = - .1f + .15fc $$

Now with this in mind lets graph these analyically. For our estimating, we will use a timestep of .3, and a total time of 10 seconds. before doing anything, if we analyze the equations, we should notice that if there were no foxes, the chickens would just grow at a rate of .1 per time step. And if there were no chickens, the foxes would eventually drop to zero at a rate of -.15 per time step. so if we start with 20 chickens and 0 foxes, we should expect to see the chicken line rise, and a straight line at zero for the foxes.

Awesome! seems likes we are on the right track. now lets see if we use the Heun method and see how close it is to our analyical solution. This time lets see if we see the foxes decrease with having 20 foxes, and 0 chickens.

It Looks like that the Heun method is really accurate, as it is validated by our analytical solution. In conclusion, the Heun method is a great tool to have with great accuracy for estimating solutions.