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2. Repeat the harvest experiments.
Repeat Exploration 1.1, Problem 2, using the Ricker function in the sardine ODE. What harvest level would provide a stable sustainable Pacific sardine population? Test whether the optimal harvest rate depends on the population IC. Are the results significantly different than when you used the logistic function?
Key words See also
Introduction to ODEs
A slope field and some solution curves for / = ysin(f + y).
Ordinary differential equations (ODEs) model many natural processes, so solutions of ODEs can be used to predict the behavior of those processes.
This chapter will investigate ODEs and initial value problems, their solutions, and their solution curves, along with some methods for finding solution formulas. Slope fields are introduced and used as guides to the behavior of solution curves. The path of ajuggler's ball and the descent of a sky diver are modeled by ODEs.
Differential equation; solution; integration; separation of variables; initial values; modeling; slope field; direction field; juggling; sky diving; free fall; parachute; gravity; Newton's second law
Chapter 1 for more on modeling, and Chapter 5 for more on models of motion.
? Differential Equations
Differential equations were first used in the seventeenth century to describe physical phenomena, such as the motion of orbiting planets or swinging pendulums. Since then they have been applied to processes, such as the growth of biological populations, the management of investment portfolios, and many other dynamical systems.
An ordinary differential equation is an equation involving an unknown function of one variable and one or more of its derivatives. For example, the ODE
— = ycost dt
is a statement about an unknown function y (the dependent variable) whose independent variable is t. To solve the ODE we need to find all the functions y(t) that satisfy the ODE (we will discuss what we mean by a solution in the next section).
? “Check” your understanding by identifying the independent and dependent variables and the order of each ODE (i.e., the highest-order derivative that appears):
^ = 2 y+ 2x dx
„ d2 z .dz„
3—j — 4— + 7 z= 4sin(2t) dt2 dt
? Solutions to Differential Equations
A function is a solution of an ODE if it yields a true statement when substituted into the equation. For example, y = 2t2 is a solution of the equation
§ = 4' <»
? Can you find another solution of ODE (1)?
Most ODEs have infinitely Actually, ODE (1) has infinitely many solutions. A single solution is
many solutions. called a particular solution. The set of all solutions is called the general
solution. For example, the general solution of ODE (1) is y = 212 + C, where C is any constant, while y = 212 + 3isa particular solution.
Solving a Differential Equation
? Solving a Differential Equation
Solving a differential equation involves finding a function, just as solving an algebraic equation involves finding a number.
An ODE such as dy/ dt = 2ty gives us information about an unknown function y in terms of its derivative(s). In your differential equations class, you’ll learn some methods for finding solutions of ODEs. The section “Finding a Solution Formula” later in this chapter also describes two techniques.
? Slope Fields
Slopes for ? = y cos t:
Each segment of a slope field is tangent at its midpoint to the solution curve through that midpoint.
One useful way to get information about solutions of an ODE is to graph them; graphs of solutions are called solution curves. For first-order ODEs, you can actually get a good idea of what solution curves look like without solving the equation! Notice that for the ODE ? = y cos t the slope of the solution curve passing through the point (t, y) is given by y cos t. Every firstorder ODE gives you direct information about the slope of the solution curve through a point, so you can visualize solution curves by drawing small line segments with the correct slopes on a grid of fixed points. With patience (or a computer), you can draw many such line segments (as in the chapter cover figure). This is called a slope field. (Some books call it a direction field.) With practice you’ll be able to imagine some of the line segments running together to make a graph. This approximates the graph of a solution to the ODE, that is, a solution curve. Figure 2.1 shows a slope field with several solution curves.
Figure 2.1: Slope field and seven solution curves for / = ycos t.
? Initial Values
We have seen that an ODE can have many solutions. In fact, the general solution formula involves an arbitrary constant. What happens if we specify that the solution must satisfy another property, such as passing through a given point? For example, all functions y = 2t2 + C are solutions of the ODE dy/ dt = 4t, but only the specific solution y = 2t2 + 3 satisfies the condition that y = 5 when t = 1. So, if we graph solution curves in the ty-plane, only the graph of the solution y = 2? + 3 goes through the point (1, 5).