Financial Simulation Modeling in Excel

ORDINARY DIFFERENTIAL EQUATIONS (ODE)

The next step to understanding partial differential equations (PDE) is to understand an ODE, or ordinary differential equation. But before we can even discuss that, let us first review “equations” in general and their “solutions.” Suppose we were given the following equation (equation A.8):

The solution(s) to equation A.8 is the value that x can take that satisfies the equality. In this example, there is only one solution, and that is x = 3. We employ a similar concept when dealing with ODEs. However, instead of trying to determine a set of values a variable can take to satisfy the equality, we are trying to find a function that, when its derivates are taken, satisfies the given differential equation. An ODE deals strictly with regular derivatives, and as we saw in the last section, this means they only operate on functions with one variable. In their simplest form, ODEs look something like equation A.9:

The solution to this equation is any function, f(x), that when its derivative with respect to x is taken, produces 4 (equation A.10).

You may notice that equation A.10 is just a rewrite of equation A.9. Ultimately the solution to this particular ...