Now in Chapter 2, we will discuss the very interesting and exciting topic so-called stability of autonomous systems of the first-order equations in the plane. In the previous chapter, Chapter 1, we mainly focused on the method of self solving linear systems of differential equations with constant coefficients. We now consider the system of nonlinear first-order differential equations, because it involves the nonlinear differential equation. Usually we cannot expect it to find the explicit solutions for the system. In most cases, it's not just difficult but it's impossible. However, without knowing the explicit solutions, we can still obtain some very important information about the solution. It is the so-called the qualitative study of differential equations. We're not going to handle the most general case, but, we confine our service to systems of two equations involving two unknowns. Two unknowns and say x of t and y of t, for two unknowns. The systems that we have, we can write it in its normal form is the x prime of t is equal to g_1 of x, y and t, and y prime of t that is equal to g_2 of x, y of q. Because g_1 and the g_2 are quiet arbitrary so in general, the system is a nonlinear system of the first-order differential equations, nonlinear system. We call this system to be autonomous if both right inside the g_1 and the g_2. If g_1 x, y, t is equal to g_1 of x y, and the g_2 of x, y of t is equal to g_2 of x y. In other words, both the g_1 and the g_2 are independent of t. In that case, we say that the system is an autonomous system, autonomous because we have only two unknowns. Sometimes it'll be recorded as a plane autonomous system. We have say, a plane autonomous system, say x prime is equal to g_1 x, y, and y prime is equal to g_2 of x, y. The independent variable t do not appear in this system explicitly but it appears only implicitly in terms of x of t and y of q, so it's hidden in some sense. For example, let's go back to the system we have handled before. We consider the following toggle problem, X prime is equal to AX plus F. I'm assuming that A is an constant matrix. If the vector F is zero, then it's homogeneous otherwise, this is a non-homogeneous. That is something we ever considered in Chapter 1. Then this is autonomous, this is autonomous if and only if F is a constant matrix. That's a simple example for this plane autonomous system. X is equal to column vector x, y. The g of X is equal to column vector g1 or g2. Then using these two matrix, we can simply write this plane autonomous system as the X of prime is equal to g of X. That's the matrix expression for this plane autonomous system. As a very simple example, consider a single second-order autonomous equation. Single second-order autonomous differential equation. Say x double prime is equal to g of x, and x of prime. Because I said this is an autonomous equation, so the independent variable t is hidden. This is second-order single autonomous equation for the unknown x. Let's say introduce one another, the variable. Depend on the variable, say y, in this form. Set y is equal to x of prime. Then you can rely to the given second-order equation. X double prime is equal to y prime. You can write this as y prime is equal to g of x and y. Together with this one. Then you can write it as x of prime is equal to y, and the y prime is equal to g of x, y. We get a good example of plane autonomous system. Which we derive from a single second-order autonomous system here. This is just the good such example. As a concrete example, we may think about the motion of the pendulum. Consider this. A mass m is hanging over to the ceiling. It has a length l. I'm assuming that the string itself has no mass at all. As an idea case, so assuming that. The maximum displacement, like this one, and this is the angle Theta of t. Here's the ceiling. Let's think about the motion of this weightless rod of length l, and the mass m well touches to it's end. Then the displacement angle Theta t, must satisfy using the simple physics. In the simple physics, you can say that it satisfies the Theta double prime of t, plus g over l, and the sine of Theta of t, and that is equal to 0. That's the governing equation for the motion of the pendulum. As I said, this is the length of the rod. What is g? G is the gravitational acceleration. We can transform this to single second-order, and this is autonomous. The independent variable t is hidden. Is a simple method to make it to be the plane autonomous system. Now, x of t is equal to Theta of t, and the y of t is equal to x prime of t. Then, said a double prime is x double prime. X double prime is y prime in fact. You can rely to this one as y prime of t, plus g over l, and the sine of x of t, and that is equal to 0. We have this. That means what? X prime is equal to y, and y prime is equal to minus g over l, and the sine of x. That's a good example of plane autonomous system.
