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1.1 Basic Models and Direction Fields

A differential equation (DE) is an equation that contains derivatives.

Examples:

\[ \frac{dy}{dt} = 2y + 3t \]\[ y' = y(4-y) \]\[ y'' + 3y' + y = 0 \]
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\(\rightarrow y = ?\)

Where do DEs come from in application?

Population:
\[ \frac{dP}{dt} = rP \]
\(P\) is population, \(r\) is rate
Interest:
\[ \frac{dm}{dt} = rm \]
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Newton's 2nd Law: \(F = ma\)

Free Fall:

Consider an object falling under the influence of gravity and air resistance (drag).

Free body diagram of a falling object with upward drag force c(dy/dt) and downward gravitational force mg.
\[ m \frac{d^2y}{dt^2} = c \frac{dy}{dt} - mg \implies y(t) \]

Substituting velocity \(v = \frac{dy}{dt}\):

\[ m \frac{dv}{dt} = cv - mg \implies v(t) \]
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Slope / Direction Field

- a way to visualize solutions qualitatively.

\[ \frac{dy}{dt} = y \]

\( \frac{dy}{dt} \) : slope of \( y(t) \)

slope of solution is equal to \( y \)

  • when \( y = 0 \), \( \frac{dy}{dt} = 0 \)
  • when \( y = 1 \), \( \frac{dy}{dt} = 1 \)

slope of \( y \) at some \( t \)

\( y' = 0 \rightarrow \) equilibrium

y does not change

Direction field for dy/dt = y showing slope segments and a green exponential solution curve passing through (0,1).
Figure 1: Direction field and solution curve for \( \frac{dy}{dt} = y \).

tangent to solution

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Example

\[ y' = 3 - 2y \]

find equilibrium: \( y' = 0 \rightarrow y = \frac{3}{2} \)

Direction field for y' = 3-2y showing slopes converging toward the horizontal equilibrium line at y = 1.5.
Figure 2: Direction field and convergence for \( y' = 3 - 2y \).

Initial Condition

  • if \( y(0) = \frac{3}{2} \)
    as \( t \to \infty, y \to \frac{3}{2} \)
  • if \( y(0) > \frac{3}{2} \)
    as \( t \to \infty, y \to \frac{3}{2} \)
  • if \( y(0) < \frac{3}{2} \)
    as \( t \to \infty, y \to \frac{3}{2} \)

all solutions converge to \( y = \frac{3}{2} \) as \( t \to \infty \)

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Example: Autonomous Differential Equation

\[ y' = y(y - 2) \]

Equilibria: \( y = 0, y = 2 \)

Direction Field Analysis

The behavior of the solution curves can be analyzed by examining the sign of \( y' \) in different regions defined by the equilibrium solutions.

Region: \( y > 2 \)

  • \( y' > 0 \)
  • The solution grows as \( y \) increases.

Region: \( 0 < y < 2 \)

  • \( y' < 0 \)
  • \( y' \) is small near \( y = 2 \) and \( y = 0 \).

Region: \( y < 0 \)

  • \( y' > 0 \)
  • The slope is steep as \( y \) decreases.
Direction field for y' = y(y-2) with horizontal lines at y=0 and y=2 . Curves diverge above y=2 and below y=0 , and converge toward y=0 between the equilibria.
Figure 1: Direction field and integral curves for the autonomous ODE.
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Software Resources

dfield8

Available from Matlab

or via the web:

http://comp.uark.edu/~aeb019/dfield.html

Note: The character in the URL is a "zero" (0).