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Trench Many physical problems concern relationships between changing quantities. Since rates of change are repre- sented mathematically by derivatives, mathematical models often involve equations relating an unknown function and one or more of its derivatives. Such equations are differential equations. They are the subject of this book. Similarly, much of this book is devoted to methods that can be applied in later courses.
In this section we mention a few such applications. The mathematical model for an applied problem is almost always simpler than the actual situation being studied, since simplifying assumptions are usually required to obtain a mathematical problem that can be solved. For example, in modeling the motion of a falling object, we might neglect air resistance and the gravitational pull of celestial bodies other than Earth, or in modeling population growth we might assume that the population grows continuously rather than in discrete steps.
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All the examples in this section deal with functions of time, which we denote by t. In the Malthusian model, it is assumed that a P is a constant, so 1. Section 1. This model assumes that the numbers of births and deaths per unit time are both proportional to the population. The constants of proportionality are the birth rate births per unit time per individual and the death rate deaths per unit time per individual ; a is the birth rate minus the death rate.
However, the model must inevitably lose validity when the pre- diction exceeds these limits. Perhaps the most famous model of this kind is the Verhulst model, where 1. Equation 1. You will learn how to solve it in Section 1. See Exer- cise 2. Figure 1. Common sense suggests this. In this second situation we must use a model that accounts for the heat exchanged between the object and the medium. Again, we assume that T and Tm are related by 1. Solving this for Tm and substituting the result into 1.
Spread of Epidemics One model for the spread of epidemics assumes that the number of people infected changes at a rate proportional to the product of the number of people already infected and the number of people who are susceptible, but not yet infected. Graphs of this function are similar to those in Figure 1. Since the second and no higher order derivative of y occurs in this equation, we say that it is a second order differential equation.
One way to model the effect of competition is to assume that the growth rate per individual of each population is reduced by an amount proportional to the other population, so 1. It can be shown Exercise The arrows indicate direction along the curves with increasing t.
P Q L Figure 1. The order of a differential equation is the order of the highest derivative that it contains.
Elementary Differential Equations and Boundary Value Problems, 11th Edition
A differential equation is an ordinary differential equation if it involves an unknown function of only one variable, or a partial differential equation if it involves partial derivatives of a function of more than one variable. Again, this is a calculus problem. Although none of these equations is written as in 1. In this case, we also say that y is a solution of 1.
Functions that satisfy a differential equation at isolated points are not interesting. The graph of a solution of a differential equation is a solution curve. Thus, any solution curve of a differential equation is an integral curve, but an integral curve need not be a solution curve. We leave it to you to verify that these functions both satisfy 1. However, 1. Example 1.
Solution Substituting 1. Therefore y is a solution of 1. The part of the graph of 1. Solution Differentiating 1. This shows that every solution of 1. On the other hand, differentiating 1. Since the constants k1, k2, Therefore Example 1. Initial Value Problems In Example 1.
The next example illustrates this.
We can rewrite the problem considered in Example 1. We call this an initial value problem. Initial value problems can also be posed for higher order differential equations. These requirements are the initial conditions. For example, we would write an initial value problem for 1. Since the function in 1. By evaluating 1.
Similarly, the interval of validity of 1. Free Fall Under Constant Gravity The term initial value problem originated in problems of motion where the independent variable is t representing elapsed time , and the initial conditions are the position and velocity of an object at the initial starting time of an experiment. Assume that gravity is the only force acting on the object. Therefore the solution of the initial value problem 1. Find the order of the equation. Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function.
Find all solutions of the equation. Solve the initial value problem.
Edwards & Penney, Elementary Differential Equations with Boundary Value Problems | Pearson
Verify that the function is a solution of the initial value problem. Let a be a nonzero real number. Identify the open interval. In this case we may resort to graphical or numerical methods to get some idea of how the solutions of the given equation behave. In Section 2. In Chapter 3 we discuss numerical methods for obtaining approximate solutions of 1. Recall that a solution of 1. Therefore, not being able to solve 1. Through each point in the grid we draw a short line segment with slope f xi, yj.
However, there is software for doing this. These methods can be used to plot solution curves of 1. Figures 1. As we saw in Example 1. The integral curves near the top and bottom are solution curves.
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However, the integral curves near the middle are more complicated. For example, Figure 1. There are three solution curves of 1. Often you may not completely understand how the software does what it does. Just be careful that you use the technology as a supplement to thought rather than a substitute for it.
Elementary Differential Equations with Boundary Value Problems: Pearson New International Edition
Sketch some integral curves. Choose a positive a and an arbitrary b. Vary T, c, and d until you discover a common property of all the solutions of C. Repeat this experiment with various choices of a and b until you can state this property precisely in terms of a and b.