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  • Draw the direction field for a given first-order differential equation.
  • Use a direction field to draw a solution curve of a first-order differential equation.
  • Use Euler’s Method to approximate the solution to a first-order differential equation.

For the rest of this chapter we will focus on various methods for solving differential equations and analyzing the behavior of the solutions. In some cases it is possible to predict properties of a solution to a differential equation without knowing the actual solution. We will also study numerical methods for solving differential equations, which can be programmed by using various computer languages or even by using a spreadsheet program, such as Microsoft Excel.

Creating direction fields

Direction fields (also called slope fields) are useful for investigating first-order differential equations. In particular, we consider a first-order differential equation of the form

y = f ( x , y ) .

An applied example of this type of differential equation appears in Newton’s law of cooling, which we will solve explicitly later in this chapter. First, though, let us create a direction field for the differential equation

T ( t ) = −0.4 ( T 72 ) .

Here T ( t ) represents the temperature (in degrees Fahrenheit) of an object at time t , and the ambient temperature is 72 ° F . [link] shows the direction field for this equation.

A graph of a direction field for the given differential equation in quadrants one and two. The arrows are pointing directly to the right at y = 72. Below that line, the arrows have increasingly positive slope as y becomes smaller. Above that line, the arrows have increasingly negative slope as y becomes larger. The arrows point to convergence at y = 72. Two solutions are drawn: one for initial temperature less than 72, and one for initial temperatures larger than 72. The upper solution is a decreasing concave up curve, approaching y = 72 as t goes to infinity. The lower solution is an increasing concave down curve, approaching y = 72 as t goes to infinity.
Direction field for the differential equation T ( t ) = −0.4 ( T 72 ) . Two solutions are plotted: one with initial temperature less than 72 ° F and the other with initial temperature greater than 72 ° F .

The idea behind a direction field is the fact that the derivative of a function evaluated at a given point is the slope of the tangent line to the graph of that function at the same point. Other examples of differential equations for which we can create a direction field include

y = 3 x + 2 y 4 y = x 2 y 2 y = 2 x + 4 y 2 .

To create a direction field, we start with the first equation: y = 3 x + 2 y 4 . We let ( x 0 , y 0 ) be any ordered pair, and we substitute these numbers into the right-hand side of the differential equation. For example, if we choose x = 1 and y = 2 , substituting into the right-hand side of the differential equation yields

y = 3 x + 2 y 4 = 3 ( 1 ) + 2 ( 2 ) 4 = 3.

This tells us that if a solution to the differential equation y = 3 x + 2 y 4 passes through the point ( 1 , 2 ) , then the slope of the solution at that point must equal 3 . To start creating the direction field, we put a short line segment at the point ( 1 , 2 ) having slope 3 . We can do this for any point in the domain of the function f ( x , y ) = 3 x + 2 y 4 , which consists of all ordered pairs ( x , y ) in 2 . Therefore any point in the Cartesian plane has a slope associated with it, assuming that a solution to the differential equation passes through that point. The direction field for the differential equation y = 3 x + 2 y 4 is shown in [link] .

A graph of the direction field for the differential equation y’ = 3 x + 2 y – 4 in all four quadrants. In quadrants two and three, the arrows point down and slightly to the right. On a diagonal line, roughly y = -x + 2, the arrows point further and further to the right, curve, and then point up above that line.
Direction field for the differential equation y = 3 x + 2 y 4 .

We can generate a direction field of this type for any differential equation of the form y = f ( x , y ) .

Definition

A direction field (slope field)    is a mathematical object used to graphically represent solutions to a first-order differential equation. At each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point.

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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