Chapter 6: Differential Equations

First, a little review: Consider: or then: It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given: find We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

If we have some more information we can find C. Given: and when , find the equation for This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

Ex: Graph a General Solution

Graph the family of functions that solve the equation: Which family of functions solves this differential equation? While it is impossible to graph all of the equations, we can graph enough to see what a family of solutions may look like. Store the following values for L1 by doing the following: Now, graph the function: y1 = sin (x) + L1

Notice that the graph shows all the “parallel” curves Notice that the graph shows all the “parallel” curves. This shouldn’t be too surprising as the functions are all vertical translations of the basic sine curve. It may be less obvious that we could have predicted the appearance of this family of curves from the differential equation itself.

Exploration 1 on page 323

This exploration shows that we could have produced the family of curves by carefully looking at the slopes. This introduces the idea behind slope fields.

Initial value problems and differential equations can be illustrated with a slope field. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator. Recent AP tests have asked students to draw one by hand.

Draw a segment with slope of 2. Draw a segment with slope of 2. 1 2 3 1 2 Draw a segment with slope of 0. 1 1 2 Draw a segment with slope of 4. 2 4 -1 -2 -2 -4

If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

Slope fields are useful because they can give you the basic shape of the general equation even when you can’t find the antiderivative analytically. By drawing a bunch of short line segments that give the slope at any point (x, y), we can approximate the solution curve at any specific point.

Solving a differential equation analytically can be difficult or impossible. But, by considering a graphical approach, we can learn volumes about the solution. A slope field shows the general shape of all solutions and can be helpful in getting a visual perspective of the directions of the solutions.

Construct a slope field for the differential equation: Ex 6 page 323

Construct a slope field for the differential equation: Then use the slope field to sketch the solution that passes through the point (1, 1).

Geometer’s Sketch Pad time

Calculator Time Ti-84s and Ti-83s can download a program from Ti’s site or google TI-83/84 and Slope Field TI-89 will be next CX- We have it easy.

Use your calculator to construct a slope field for the equation: Then sketch the graph of the equation that passes through the point (2, 0) Use the inspire software to graph this. You will need to type in y1 for the y-value*

For more challenging differential equations, we will use the calculator to draw the slope field. On the TI-89: Push MODE a and change the Graph type to DIFF EQUATIONS. MODE Go to: Y= Press and make sure FIELDS is set to SLPFLD. I Go to: and enter the equation as: Y= (Notice that we have to replace x with t , and y with y1.) (Leave y1 blank.)

Set the viewing window: WINDOW Then draw the graph: GRAPH

Be sure to change the Graph type back to FUNCTION when you are done graphing slope fields.

Yo’ Turn…