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Math 3120 Differential Equations with Boundary

Value Problems

Chapter 2 First-Order Differential

Equations Section 2-1 Solution Curves without

a solution

Methods to find solutions of DE

- Analytical Methods Look for explicit formulas to

describe the solution of a DE. For example,

separable variable, exact equations, etc. - Qualitative Methods Geometric techniques for

representing equations. For example, Direction

fields, phase line, phase portraits. - Numerical Methods Approximates solutions to IVP

using calculator or computers. For example, Euler

method, Runge-Kutta method, etc.

Direction Field (DF) / Slope fields

- Represent the motion of an object falling in the

atmosphere near sea level by the DE - Look at the qualitative aspects of solution of

Eq. (1) without actually solving them. - Taking g 9.8 m/sec2, m 10 kg, ? 2 kg/sec,
- we obtain
- Investigate the behavior of the solutions of Eq.

(2) without solving the DE.

Example 1 Sketching Direction Field

- Using differential equation and table, plot

slopes (estimates) on axes below. The resulting

graph is called a direction field. (Note that

values of v do not depend on t.)

Direction Field Equilibrium Solution

- Arrows give tangent lines to solution curves, and

indicate where soln is increasing decreasing

(and by how much). - Horizontal solution curves are called equilibrium

solutions. - Use the graph below to solve for equilibrium

solution, and then determine analytically by

setting v' 0.

Note

- Visually the DF suggest the appearance or shape

of family of solution curves to the DE. - Each line segment is tangent to the graph of the

solutions of DE that passes through that point. - Use these line segments as a guide to sketch the

graph of the solutions of the DE. - Look at the Long-term behavior of solutions of

the DE. - Use Mathematica to generate a DF.

Sketch some Solution Curves

- When graphing direction fields, be sure to use an

appropriate window, in order to display all

equilibrium solutions and relevant solution

behavior.

Equilibrium Solutions

- In general, for a differential equation of the

form - find equilibrium solutions by setting y' 0 and

solving for y - Example Find the equilibrium solutions of the

following.

Example 2 Graphical Analysis

- Discuss solution behavior as t ? 8 for the

differential equation below, using the

corresponding direction field.

Example 3 Graphical Analysis

- Discuss how the solutions behave as t ? 8 for the

differential equation below, using the

corresponding direction field.

Example 4 Graphical Analysis for a Nonlinear

Equation

- Discuss solution behavior and dependence on the

initial value y(0) for the differential equation

below, using the corresponding direction field.

Autonomous First-Order Differential Equations

- In this section, we examine equations of the form
- called autonomous equations, where the

independent variable t does not appear

explicitly. - For Example
- A real number c is called a Critical Point (CP)

of the function y' f (y) if it is a zero of f,

i.e., f( c) 0. Critical points are also called

equilibrium or stationary points. - Many models of physical laws have DE that are

autonomous. For instance, Exponential Growth,

Logistic Growth, Newtons Law of Cooling etc.

Phase Lines

- Autonomous equations have slope field where the

lineal elements (small tangent lines) are

parallel along horizontal line in the t-y plane,

i.e. two points with the same y-coordinate but

different t-coordinate have the same lineal

elements. - Thus, there is great deal of redundancy in the

slope field of autonomous equations. Thus, we

need to draw one vertical line containing the

information. This line is called the phase line

for the autonomous equation.

Phase Portrait

- Phase Portraits is a phase line marked by a point

where the derivative is zero and also indicate

the sign of the derivative on the intervals in

between. - The phase portraits provide qualitative

information about the solutions of the

differential equation

How to Draw Phase Portraits

- For the autonomous equation.
- Draw the y-line ( of the t y-plane).
- Find the equilibrium points ( numbers given by

f(y)0), and mark them on the line. - Find the intervals of y-values for which f(y) gt

0, and draw arrows pointing up these intervals. - Find the intervals of y-values for which f(y) lt

0, and draw arrows pointing down these intervals. - Example Draw the phase portrait of

Attractors and Repellers

- Let y(t) be a non-constant solution of the

autonomous DE given by y' f(y) and c be the

critical point of the DE. - If all solutions y(t) of (3) that start from an

initial point (x0,y0) sufficiently near c exhibit

the asymptotic behavior limx?8 y(t) c, then c is

said to be asymptotically stable. Asymptotically

stable critical points are also called attractors

. - If all solutions y(t) of (3) that start from an

initial point (x0,y0) move away from c as t

increases, then c is said to be unstable .

Unstable critical points are also called

repellers. - If c exhibits characteristic of both an attractor

and a repeller, then say that the critical point

c is semi-stable.

Example 5 CP, Phase Portraits

- Find the critical points, phase portrait of the

given autonomous DE. Classify each critical

points as asymptotically stable, unstable, or

semi-stable.

How to use Phase Portraits to sketch Solutions

- Consider the Differential Equations

Summarize

- Let y(t) be a non-constant solution of the

autonomous DE given by y' f(y) on a Region R

corresponding to an interval I and c1, c2 be the

critical points of the DE such that c1lt c2 . Then

the graph of the equilibrium solutions partition

the region r into R1, R2, R3. The following

conclusions can be drawn - y(t) is a solution that passes through (t0,y0)

in a sub region Ri ,then y(t) remains in that

region for all t. - f(y) cannot change signs in a sub region.
- Since y' is either positive or negative in Ri ,

therefore y(t) is strictly monotonic. - If y(t) is bounded above by a critical point ci ,

then the graph of y(t) must approach the graph of

the equilibrium solutions either as t?8 or t?-8 . - If y(t) is bounded below by a critical point ci ,

then the graph of y(t) must approach the graph of

the equilibrium solutions either as t?8 or t?-8 . - If y(t) is bounded above and below by two

consecutive critical points ci , then the graph

of y(t) must approach the graph of both the

equilibrium solutions , one as t?8 and the other

as t?-8 .

Example 6 Phase Portraits, Solution Curves

- Sketch the phase portrait and solution curves

for the following differential equation.

Example 7 Qualitative Analysis for the Logistic

Population Model

- Logistic Population Model states that the rate of

growth of the population is proportional to the

size of the population with the assumptions that

there is limitation of space and resources. - Variables time t, population P
- Parameter growth rate coefficient k, carrying

capacity N. - The Logistic Population Model is given by

Warning Not all Solutions exist for all time

- Sketch the phase portrait and solution curves

for the following differential equation. - Note We cannot tell if solutions will blow up in

finite time by simply looking at the phase line.