Jeremy Côté

Familiar Forms

When you first start solving a problem in mathematics, the goal is often to find a way to express the problem as some sort of differential equation. During this initial search, you don’t care how the equation looks. It’s more important to get it written down so that you can proceed.

However, once you do have an equation, the first step is not to try and solve it. That’s a rookie mistake. Instead, the question you should be asking yourself is, “Can I put this equation into a form I recognize?” Asking yourself this question can save a ton of time in solving a problem. After all, if you can recognize the form of the equation, then you know the answer without doing any more work.

This might seem like an edge case that never happens in practice, but that’s not true. In particular, mathematicians have studied the solutions of many ordinary and partial differential equations, and know the answers. Therefore, if you’re working with a differential equation (which is almost always the case in physics), you might be able to save yourself a lot of time if you recognize the form of the equation.

For example, any student in physics who has taken more than a few courses will recognize the differential equation representing simple or damped harmonic motion. Physics students come across it all the time. This equation comes up when considering swinging pendulums, motion of a spring, electrical circuits, stability of circular orbits, and even in the Schrödinger equation. It’s what you might call a pervasive equation.

I can guarantee that professors don’t go over the solution to this equation after perhaps the first semester of physics. The reason is that students learn how to solve this differential equation, so there’s no need to go through all the work again and again. Instead, they identify the equation, and then give the solution.

However, it might not always be obvious that an equation satisfies the differential equation for harmonic motion. If there are a bunch of constants littered everywhere in the equation (due to the physical situation), it can be difficult to see the underlying equation. How do we deal with this so that we can try and identify an equation?

The trick is to change variables and bundle up constants together as much as you can. If your equation has constants littered everywhere, see if you can divide the constants out so that you have less in total. In the same vein, if you can see a simple change of variables that will allow you to “absorb” some of your constants in the differential equation, that can also help in simplifying the equation.

The goal here is to try to make your equation as generic as possible. That’s often the best way to compare it to the known equations in mathematics which have solutions. When you look at solutions to differential equations, they won’t be given in terms of parameters like the mass of a particle. The constants will be generic. Therefore, it’s often in one’s best interest to “clean up” a differential equation as soon as possible in order to make it recognizable.

Remember, there’s nothing wrong with ploughing ahead and solving the equation right off. It can still work. It’s just that the constants present in an equation that are specific to the problem can muddy the waters of the solution. By dividing constants out and changing variables, the equation will shed its “particular” qualities, showing only the essence underneath. Then, one can save time by identifying it with a known differential equation.

The point when solving a physical problem isn’t to go through all of the mathematical detail for no reason. If a solution is already known, there’s no point to ignore that. Use the fact that you can recognize solutions to speed up your problem solving. In the end, it’s the physical solution itself that matters.