Oscilations part 1

Linear differential equations

A linear differential equation is one in which x and its time derivatives enter only through their first powers. An example is 3x¨+7x˙ +x = 0. An example of a nonlinear differential equation is 3x¨ + 7x˙² + x = 0.

If the right-hand side of the equation is zero, then we use the term homogeneous differential equation. If the right-hand side is some function of t, as in the case of 3x¨ - 4x˙ = 9t² - 5, then we use the term inhomogeneous differential equation. The goal of this chapter is to learn how to solve these two types of equations. Linear differential equations come up again and again in physics, so we had better find a systematic method of solving them.

The techniques that we will use are best learned through examples, so let’s solve a few differential equations, starting with some simple ones. Throughout this chapter, x will be understood to be a function of t. Hence, a dot will denote time differentiation.

Example 1 (x˙ = ax): This is a very simple differential equation. There are two ways (at least) to solve it.

First method: Separate variables to obtain dx/x = adt, and then integrate to obtain lnx = at + c. Exponentiate to obtain

x = Ae^at, (3.1)

where A = e^c is a constant factor. A is determined by the value of x at, say, t = 0.

Second method: Guess an exponential solution, that is, one of the form x = Ae^at. Substitution into x = ax immediately gives a = a. Therefore, the solution is x = Ae^at. Note that we can’t solve for A, due to the fact that our differential equation is homogeneous and linear in x (translation: A cancels out). A is determined from the initial condition.

This method may seem a bit silly, and somewhat cheap. But as we will see below, guessing these exponential functions (or sums of them) is actually the most general thing we can try, so the method is indeed quite general.

Remark: Using this method, you may be concerned that although we have found one solution, we might have missed another one. But the general theory of differential equations says that a first-order linear equation has only one independent solution (we’ll just accept this fact here). So if we find one solution, then we know that we’ve found the whole thing. *

Example 2 (x = ax): If a is negative, then this equation describes the oscillatory motion of, say, a spring. If a is positive, then it describes exponentially growing or decaying motion. There are two ways (at least) to solve this equation.

First method: We can use the separation-of-variables method of Section 2.3 here, because our system is one in which the force depends on only the position x. But this method is rather cumbersome, as you found if you did Exercise 2.10 or 2.11. It will certainly work, but in the case where our equation is a linear function of x, there is a much simpler method:

Second method: As in the first example above, we can guess a solution of the form x(t) = Ae^at and then find out what a must be. Again, we can’t solve for A, because it cancels out. Plugging Ae^at into i¨ = ax gives a = ±^. We have therefore found two solutions. The most general solution is an arbitrary linear combination of these,

x(t) = Ae^¹ + Be-^\ (3.2)

as you can quickly check. A and B are determined from the initial conditions.

Very Important Remark: The fact that the sum of two different solutions is again a so­lution to our equation is a monumentally important property of linear differential equations. This property does not hold for nonlinear differential equations, for example x² = x, because the act of squaring after adding the two solutions produces a cross term which destroys the equality, as you should check.

This property is called the principle of superposition. That is, superimposing two solutions yields another solution. This quality makes theories in physics that are governed by linear equations much easier to deal with than those that are governed by nonlinear ones. General Relativity, for example, is permeated with nonlinear equations, and solutions to most General Relativity systems are extremely difficult to come by.