Wait, Change is Better Than Quantity?
When we think about the math that we have learned throughout school, the overwhelming majority deals with absolute quantities. For example, if given the linear function f(x) = x + 2, for whatever input x, we obtain an output that is a fixed number quantity: if x = 2, f(x) = 4, and if x = 5.5, f(x) = 7.5. For the longest time, I thought that this was all that math entailed. That was until I first learned about calculus.
In calculus, the derivative was introduced, opening up the world of representing rate of change. I was surprised to see how useful this rate of change was: it could tell us where the function was increasing, decreasing, or if it was at a maximum or minimum. However, as we moved to differential equations, I became the most intrigued. At the heart of it, differential equations equate a function to its derivative. Instead of f(x) = … we have dy/dx = … Differential equations are very important in the world of modeling, as multiple laws from physics, economics, and chemistry can be represented by these types of equations. The heat equation in physics and the Black-Scholes equation in finance are just a few famous models.
In this article, I’ll start with a brief introduction to differential equations, including methods for separable differential equations and first order linear differential equations.
Separable Differential Equations
As mentioned before, differential equations talk in terms of change rather than quantity. Therefore, when we solve a differential equation, our answer is in terms of a function. In fact, there are infinitely many answers to basic differential equations, as we are simply finding an antiderivative in most cases. The first type of differential equation we’ll consider are Separable Differential Equations.
Consider the problem below.
We are given information about the derivative of the function y, and we need to find the original function. How should we go about solving this? The name “Separable Differential Equation” should give us a subtle hint about our preferred method of tackling this problem: separate the equation such that all y’s are on one side and all x’s are on the other. We…