Figure 1: Few solutions of
It turns out that solving differential equations can be quite hard. There is no general method that solves every differential equation. We will generally focus on how to get exact formulas for solutions of certain differential equations, but we will also spend a little bit of time on getting approximate solutions.
For most of the course we will look at ordinary differential equations or ODEs. by which we mean that there is only one independent variable and derivatives are only with respect to this one variable. If there are several independent variables, we will get partial differential equations or PDEs. We will briefly see these near the end of the course.
Even for ODEs, which are very well understood, it is not a simple question of turning a crank to get answers. It is important to know when it is easy to find solutions and how to do so. Although in real applications you will leave much of the actual calculations to computers, you need to understand what they are doing. It is often necessary to simplify or transform your equations into something that a computer can understand and solve. You may need to make certain assumptions and changes in your model to achieve this.
To be a successful engineer or scientist, you will be required to solve problems in your job that you have never seen before. It is important to learn problem solving techniques, so that you may apply those techniques to new problems. A common mistake is to expect
to learn some prescription for solving all the problems you will encounter in your later career. This course is no exception.
0.2.3 Differential equations in practice
So how do we use differential equations in science and engineering? First, we have some real world problem we wish to understand. We make some simplifying assumptions and create a mathematical model . That is, we translate the real world situation into a set of differential equations. Then we apply mathematics to get some sort of a mathematical solution . There is still something left to do. We have to interpret the results. We have to figure out what the mathematical solution says about the real world problem we started with.
Learning how to formulate the mathematical model and how to interpret the results is what your physics and engineering classes do. In this course we will focus mostly on the mathematical analysis. Sometimes we will work with simple real world examples, so that we have some intuition and motivation about what we are doing.
Let us look at an example of this process. One of the most basic differential equations is the standard exponential growth model . Let
denote the population of some bacteria on a Petri dish. We assume that there is enough food and enough space. Then the rate of growth of bacteria is proportional to the population—a large population grows quicker. Let
denote time (say in seconds) and
the population. Our model is