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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