Ordinary Differential Equations

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Ordinary Differential Equations

• CHAPTER 1

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Contents

• 1.1 Definitions and Terminology
• 1.2 Initial-Value Problems
• 1.3 Differential Equations as Mathematical Models

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1.1 Definitions and Terminology

• Introduction differential equations means that
  equations contain derivatives, eg dy/dx
  0.2xy (1)
• Ordinary DE An eq. contains only ordinary
  derivates of one or more dependent variables of a
  single independent variable. eg dy/dx 5y
  ex, (dx/dt) (dy/dt) 2x y (2)

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• Partial DE An equation contains partial
  derivates of one or more dependent variables of
  two or more independent variables.
  
  (3)
• Notations Leibniz notation dy/dx, d2y/ dx2
  prime notation y, y, ..
  Subscript notation ux, uy, uxx, uyy, uxy , .
• Order highest order of derivatives

first order
second order
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• General form of n-th order ODE
  
  (4)
• Normal form of (4)
  
  (5)eg normal form of 4xy y x, is
  y (x y)/4x
• Linearity An n-th order ODE is linear if F is
  linear in y, y, y, , y(n). It means when (4)
  is linear, we have
  (6)

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• The following cases are for n 1 and n 2
  and
  (7)
• Two properties of a linear ODE1) y, y, y,
  are of the first degree.2) Coefficients a0, a1,
  , depend at most on x
• Nonlinear examples

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• That is, a solution of (4) is a function ?
  possesses at least n derivatives and F(x, ?(x),
  ?(x), , ?(n)(x)) 0 for all x in I, where I is
  the interval ? is defined on.

DEFINITION 1.2
Solution of ODE

• Any function ?, defined on an interval I,
  possessing at least n derivatives that are
  continuous on I, when replaced into an n-th
  order ODE, reduces the equation into an
  identity, is said to be a solution of the
  equation on I.

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Example 1

• Verify the indicated function is a solution of
  the given ODE on (-?, ?) (a) dy/dx xy1/2 y
  x4/16 (b)
• Solution (a) Left-hand side Right-hand
  side then left right
• (b) Left-hand side Right-hand side 0
  then left right

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• Suppose y 0 is a solution of ODE, then y 0 is
  called a trivial solution

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Example 2 Function vs Solution

• y 1/x, is the solution of xy y 0, however,
  this function is not differentiable at x 0. So,
  the interval of definition I is (-?, 0), or (0,
  ?).

Fig 1.1
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• Explicit solution dependent variable is
  expressed solely in terms of independent variable
  and constants.Eg solution is y ?(x).

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Example 3

• x2 y2 25 is an implicit solution of
  dy/dx -x/y (8)on the interval -5 lt x lt 5.
• Since dx2/dx dy2/dx (d/dx)(25)then 2x
  2y(dy/dx) 0 and dy/dx -x/ysolution curve is
  shown in Fig1.2

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Fig1.2

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• Families of solutions A solution containing an
  arbitrary constant c is called a one-parameter
  family of solutions. A solution containing n
  arbitrary constants c1, c2, , cn is called a
  n-parameter family of solutions.
• Particular solution A solution free of arbitrary
  parameters. eg y cx x cos x is a solution
  of xy y x2 sin x on (-?, ?), y x cos x
  is a particular solution corresponding to c 0.
  See Fig1.3

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Fig1.3

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Example 4

• x c1cos 4t and x c2 sin 4t are solutions of
  x? 16x 0we can easily verify that x
  c1cos 4t c2 sin 4t is also a solution.

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Example 5

• We can verify y cx4 is a solution of xy? 4y
  0 on (-?, ?). See Fig1.4(a).
• The piecewise-defined function is a
  particular solution where we choose c -1 for x
  lt 0 and c 1 for x ? 0. See Fig1.4(b).

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Fig1.4

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• Singular solution A solution can not be obtained
  by particularly setting any parameters. y
  (x2/4 c)2 is the family solution of dy/dx
  xy1/2 , however, y 0 is also a solution of the
  above DE.
• We cannot set any value of c to obtain the
  solution y 0, so we call y 0 is a singular
  solution.

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• System of DEs two or more equations involving
  two or more unknown functions of a single
  independent variable. dx/dt f(t, x, y) dy/dt
  g(t, x, y) (9)

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1.2 Initial-value Problems

• Introduction We are often interested in a
  solution y(x) of a DE satisfying an initial
  condition.
• Example on some interval I containing xo,
  solve subject to
  (1)This is called an Initial-Value Problem
  (IVP).
• y(xo) yo , y?(xo) y1 ,are called initial
  conditions.

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• First and Second Order IVPs (2)and
  (3)are first and second order
  initial-value problems, respectively. See Fig1.7
  and 1.8.

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Fig1.7 Fig1.8

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Example 1

• We know y cex is the solutions of y y on
  (-?, ?). If y(0) 3, then 3 ce0 c. Thus y
  3ex is a solution of this initial-value
  problem.If we want a solution passing through
  (1, -2), that is y(1) -2, then -2 ce, or c
  -2e-1. See Fig1.9

Fig1.9
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Example 2

• In problem 6 of sec 2.2, we have the solution of
  y 2xy2 0 to be y 1/(x2 c). If we impose
  y(0) -1, it gives c -1. Consider the
  following distinctions.
• 1) As a function, the domain of y 1/(x2 - 1) is
  the set of all real numbers except -1 and 1. See
  Fig1.10(a).
• 2) As a solution, the intervals of definition are
  (-?, -1) or (-1, 1) or (1, ?)
• 3) As a initial-value problem, y(0) -1, the
  interval of definition is (-1, 1). See Fig1.10

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Fig1.10
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Example 3

• In example 4 of Sec 1.1, we saw x c1cos 4t
  c2sin 4t is a solution of x? 16x
  0Find a solution of the following IVP x? 16x
  0, x(?/2) -2, x?(?/2) 1 (4)Solution
• Substitute x(?/2) - 2 into x c1cos 4t c2sin
  4t, we find c1 -2. In the same manner, from
  x?(?/2) 1 we have c2 ¼.

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• Existence and Uniqueness Does a solution of
  the IVP exist? If a solution exists, is it
  unique?

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Example 4

• Since y x4/16 and y 0 satisfy the DEdy/dx
  xy1/2 , and also initial-value y(0) 0, this DE
  has at least two solutions, See Fig1.11

Fig1.11
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THEOREM 1.1
Existence of a Unique Solution
Let R be the region defined by a ? x ? b, c ? y ?
d that contains the point (xo, yo) in its
interior. If f(x, y) and ?f/?y are continuous in
R, then there exists some interval Io xo- h lt
x lt xo h, h gt 0, contained in a ? x ? b and a
unique function y(x) defined on Io that is a
solution of the IVP (2).
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Fig1.12

• The geometry of Theorem 1.1 shows in Fig1.12
  

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Example 5

• For the DE dy/dx xy1/2 , we inspect the
  functions and find they are continuous in y gt
  0. From Theorem 1.1, we conclude that through any
  point (xo, yo), yo gt 0, there is some interval
  centered at xo on which this DE has a unique
  solution.

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• Interval of Existence and UniquenessSuppose y(x)
  is a solution of IVP (2), the following sets may
  not be the same
• the domain of y(x), the interval of definition
  of y(x) as a solution, the interval Io of
  existence and uniqueness.

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1.3 DEs as Mathematical Models

• Introduction Mathematical models are
  mathematical descriptions of something.
• Level of resolutionMake some reasonable
  assumptions about the system.
• The steps of modeling process are as following

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Assumptions
Mathematics formulation
Express assumptions in terms of differential
equations
If necessary, alter assumptions or increase
resolution of the model
Solve the DEs
Obtain solution
Check model Predictions with known facts
Display model predictions, e.g., graphically
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• Example Spring/mass system
• Observed from experiment damping force
  velocity
• By Newtons Law

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• Series CircuitsLook at Fig1.21.From Kirchhoffs
  second law, we have (11)where q(t) is
  the charge and dq(t)/dt i(t), which is the
  current.

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Fig1.21
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Thank You !