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— = ó cost 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 Ó 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):
-ã- = 2 Ó+ 2x dx
d2 z dz
3—r — 4— + 7z= 4sin(2f) 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
? 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 = 2t2 + C, where C is any constant, while y = 2t2 + 3 is a 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 Ó = ycos t:
Each segment of a slope field is tangent at its midpoint to
the solution curve through that
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 / = ycos t the slope of the solution curve passing through the point ( t, y) is given by ycos t. Every first-order 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 = 2t2 + 3 goes through the point (1, 5).
The requirement that y( 1) = 5 is an example of an initial condition, and the combination of the ODE and an initial condition
ft=4t, 7(1) = 5 (2)
is called an initial value problem (IVP). Its solution is y = 2²2 + 3.
? Replace the condition y(1) = 5 in IVP (2) by y(2) = 3 and find the solution of this new initial value problem. How many solutions are there?
¦ Finding a Solution Formula
An ODE usually has many solutions. How can you find a solution, and how can you describe it? A solution formula provides a useful description, but graphs and tables generated by ODE Architect are also useful, especially in the all-too-frequent case where no formula can be found. Two techniques to find solution formulas are summarized here, and others are in your textbook.