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Math 240: Transition to Advanced Math

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Title: Math 240: Transition to Advanced Math


1
Math 3120 Differential Equations with Boundary
Value Problems
Chapter 2 First-Order Differential
Equations Section 2-1 Solution Curves without
a solution
2
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.

3
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.

4
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.)

5
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.

6
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.

7
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.

8
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.

9
Example 2 Graphical Analysis
  • Discuss solution behavior as t ? 8 for the
    differential equation below, using the
    corresponding direction field.

10
Example 3 Graphical Analysis
  • Discuss how the solutions behave as t ? 8 for the
    differential equation below, using the
    corresponding direction field.

11
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.

12
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.

13
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.

14
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

15
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

16
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.

17
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.

18
How to use Phase Portraits to sketch Solutions
  • Consider the Differential Equations

19
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 .

20
Example 6 Phase Portraits, Solution Curves
  • Sketch the phase portrait and solution curves
    for the following differential equation.

21
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

22
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.
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