Slope Fields Greg Kelly, Hanford High School, Richland, Washington Adapted by Bannon, Siena College

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.

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.

1 2 3 1 2 1 1 2 2 4 -1 -2 -2 -4 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.

For more challenging differential equations, we will use the calculator to draw the slope field. On the TI-89: Push MODE 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 yi1 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.

Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Multiply both sides by dx and divide both sides by y2 to separate the variables. (Assume y2 is never zero.)

Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Combined constants of integration

Example 9: Separable differential equation Combined constants of integration

Example 9: We now have y as an implicit function of x. We can find y as an explicit function of x by taking the tangent of both sides. Notice that we can not factor out the constant C, because the distributive property does not work with tangent.