Question

[10pts] Let's imagine that we have a first-order differential equation that is hard or impossible to solve. The general form is: df g(e) f(t)-he) dt where g(t) and h(t) are understood to be known. It turns out that any first order differential equation is relatively easy to solve using computational techniques. Specifically, starting from the definition of the derivative... df f(t+dt)-S(t) (dt small) dt dt we can rearrange the equation to become... www f(t+dt)-f(t)+dt-df (dt small) dt In other words, if you know the function and its derivative at any time t, and if you choose a time interval dt that is small enough, then you can find the value of the function at the next instant of time (t+ dt). In this case, because g(t) and h(t) are known, we can easily rearrange the differential equation to be in the form, df/dt [functions of time].. Thus, the derivative is known at any time t, as long as ft is known at time t. Therefore, set up a script to numerically solve a differential equation of this type. For now, we will use g(t) 5, h(t)--6, and f(0) 3. Plot the resulting f[t) from t 0 until t 1, using at least 1000 points. Add appropriate axis labels and a meaningful title, and adjust the line thickness and font sizes. As a check, solve the differential equation analytically, and compare it to your numerical solution. This should be done robustly, i.e, the parameters of the analytic solution should depend on what's given (g, h, and f(0)). Then, construct a function that you believe should be an array of zeros, and plot it on the same axes as the numerical solution. Make the legend meaningful, e.g., it could say "check (should be zero)".