## Lecture 5

By Giuseppe Tenti and annotated by Douglas Wilhelm Harder.

We will now look at 2nd-order differential equations. A general 2nd-order differential equation is of the form

;

however, this cannot be solved in general. We will begin by looking at specific cases which have a much simpler form.

# Linear Homogeneous 2nd-order Ordinary Differential Equations with Constant Coefficients

A linear homogeneous 2nd-order differential equation with constant coefficients is a differential equation of the form:

Equation 5.1.

where , , and are arbitrary constant complex (though usually real) numbers.

This ODE is fundamental as a description of oscillation and vibration in all areas of science including electrical, mechanical, but also biological systems. To solve this, we will first look at the statements made by Leonhard Euler in 1739 in a letter to Johann Bernoulli (paraphrased):

Look, is not known, so we cannot do anything about the term in the differential equation. Therefore, the only way to make the left hand side of the equation vanish is to reduce to something like and to something like , so that the differential equation becomes something like

and then we choose and so that .

Euler then noticed that this requires that and its derivatives must be scalar multiples of each other and the only function where this is possible are the eigenfunctions of the differential operator, namely, , where and are arbitrary constants.

Under this assumption, we have that and and substituting these into Equation 5.1 yields

and therefore

and therefore, as the exponential function is not zero, we must require that

.

This is called the characteristic equation of the differential equation and is a quadratic equation with roots

.

There are three cases on the descriminant which is either positive, zero, or negative yielding two distinct real roots, a double root, or two complex conjugate roots.

## Case 1:

There are two roots and and each is a linearly independent solution ( is not constant). It follows that, by the superposition principle, the general solution is

.

#### Example 1

Find the general solution of the differential equation .

First, the characteristic equation is which has two roots and therefore and . These two solutions are linearly independent as which is not constant. Therefore, the general solution is

.

Figure 1 shows a number of solutions to this differential equation using various coefficients of and .

Figure 1. Numerous solutions to the differential equation of Example 1.

Looking at these, we note that there appear to be two classes of solutions: there is either zero or one local extreme points and all solutions tend to zero as . These three cases are shown in Figure 2.

Figure 2. Three representative examples of solutions to Example 1.

## Case 2:

In this case, the two roots are real and coincident: . This gives us one particular solution and therefore there must be a second solution which is linearly independent from this solution.

The trick for finding the second solution was first proposed by D'Alambert in 1748. He suggested that you have two solutions and where is arbitrarily small; therefore, . Consider first the two solutions with and and only once you have the two solutions, let .

Now, the two linearly independent solutions are

and .

But any linear combination of these two solutions must also be a solution and in particular, let us define

.

This must be a solution for any any finite value of but in the limit, the solution is in an indeterminate form and therefore we must use l'Hôpital's Rule:

.

The simplest form of the two linearly independent solutions is therefore

.

#### Example 2

Find the general solution of the differential equation .

First, the characteristic equation is which has a double root at . The two linearly independent solutions are, therefore, and ; hence, the general solution is

.

Figure 3 shows a number of solutions to this differential equation using various coefficients of and .

Figure 3. Numerous solutions to the differential equation of Example 2.

As with the previous case, there are either zero or one local extreme points and all solutions tend to zero as . These are the same three cases already shown in Figure 2.

## Case 3:

In this case, the roots of the characteristic equation are complex conjugates of the form where and . The two linearly independent solutions using these functions are

and ;

however, these are complex-valued functions, yet any linear combination of these solutions is also a solution. This freedom allows us to find two real-valued linearly independent solutions:

• Sum the solutions:
• Take the difference:

The constants and may be ignored and hence we may conclude that two real-valued linearly independent solutions are and . The superposition principle gives us that the general solution is

.

As an alternate formulation, we may write this using two constants in the amplitude-phase form:

where is the amplitude and is the phase shift. This can also, of course, be written in terms of a sine function together with the corresponding adjusted phase shift.

#### Example 3

Find the general solution of the differential equation .

First, the characteristic equation is which has two complex roots at . The two linearly independent solutions are, therefore, and ; hence, the general solution is

The two linearly independent solutions are, therefore, and ; hence, the general solution is

,

or, an alternate formulation is .

Figure 4 shows a number of solutions to this differential equation using various coefficients of and while Figure 5 shows the two linearly independent solutions. All solutions oscillate infinitely many times.

Figure 4. Numerous solutions to the differential equation of Example 3.

Figure 5. The two linearly independent solutions to the differential equation of Example 3.

## Examples with Animations

The reader may wonder about the transition from real to complex roots of this system, and therefore, the three animations shown in Figures 6-8 will show how the solutions to a differential equation change as the parameters change. In all cases, the differential equation will start with two different real roots, will make the transition to two identical real roots with the differential equation

and then continue with two complex roots. In each case, the reader will note that there is a smooth transition from a solution with no extreme points past to a solution which is oscillating.

Figure 6. Solutions to the differential equation

for values of .

Figure 7. Solutions to the differential equation

for values of .

Figure 8. Solutions to the differential equation for values of .

Another possibility is the transition from a solution with no roots past , to a solution with one real root past zero, to a solution which is oscillating. There do not exist solutions where there are two extreme points:

• The solution will decay towards the zero solution,
• The solution may pass the zero solution once and then decay towards it, or
• The solution will oscillate infinitely many times around the zero solution.

This natural and smooth transition from one type of solution to another should be expected, for example, calculating the integral is a polynomial for almost all values of except when of in which case the integral is a logarithm, and yet, there is a smooth transition from one state to the other, as is shown in Figure 9.

Figure 9. The integral for values of .