MATH 312 Section 2.1: Solution Curves without a Solution

Solution Curves and Direction Fields Autonomous Differential Equations Solution Curves for Autonomous DEs Conclusion

MATH 312 Section 2.1: Solution Curves without a Solution

Prof. Jonathan Duncan

Walla Walla College

Spring Quarter, 2007

Solution Curves and Direction Fields Autonomous Differential Equations Solution Curves for Autonomous DEs Conclusion

Outline

1 Solution Curves and Direction Fields 2 Autonomous Differential Equations 3 Solution Curves for Autonomous DEs 4 Conclusion

Solution Curves and Direction Fields Autonomous Differential Equations Solution Curves for Autonomous DEs Conclusion

Solving Differential Equations

In chapter 2 we start examining methods for solving differential equations. In this first section, we look at several graphical methods for finding solution curves.

Example

Consider

the

first

order

differential

equation

dy dx

=

1 - xy .

Our

geometric interpretation of the derivative tells us that if y is a

solution to this DE, then the slope of a line tangent to y at a point

(x0, y0)

is

dy dx

=

1 - x0y0.

Sketch

the

slope

of

solution

curves

which

pass through the integer coordinates (n, m) with 0 n, m 3.

Direction Field

The direction field for a first order differential equation is a graph in which each point is assigned a value equal to the slope of a solution curve at that point. These are directed line segments called lineal elements.

Solution Curves and Direction Fields Autonomous Differential Equations Solution Curves for Autonomous DEs Conclusion

Direction Field Examples

It is extremely tedious to construct direction field by hand. Instead, we often use the computer to assist.

Direction

Field

for

dy dx

=

1 - xy

Solution Curves and Direction Fields Autonomous Differential Equations Solution Curves for Autonomous DEs Conclusion

Direction Fields and IVPs

Direction fields can also be used to sketch the particular solution to an initial value problem. Example Sketch a solution curve to the initial value problem:

dy = sin(y ) cos(y )

dx subject to the constraints:

1 y (0) = 1 2 y (1) = 0

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