Transcript Differential Equations Also known as Engineering Analysis or ENGIANA Introduction In science, engineering, economics, and in most areas having a quantitative component, we are interested in describing how.

Differential Equations
Also known as
Engineering Analysis
or
ENGIANA
Introduction
In science, engineering, economics, and in
most areas having a quantitative
component, we are interested in describing
how systems evolve in time, that is, in
describing a system’s dynamics.
Introduction
• In this time series plot of a
generic state function u =
u(t) for a system, one can
monitor the state of a system
(e.g. population,
concentration, temperature,
etc) as a function of time.
• However, such curves or
formulas tell us how a
system behaves in time, but
they do not give us insight
into why a system behaves
in the way we observe.
Introduction
• Thus, we try to formulate explanatory
models that underpin the understanding we
seek.
• Often these models are dynamic equations
that relate the state u(t) to its rates of
change, as expressed by its derivatives u′(t),
u′′(t), ..., and so on.
Introduction
Before we present the formal definition, let’s
see some examples of differential equations:
g
' '
sin  0
l
1
Rq' q  sint
C
p

p'  rp 1  
K

m x' '   x
Introduction
g
' ' sin  0
L
The first equation models the angular
deflections θ = θ(t) of a pendulum of
length L.
Introduction
1
Rq' q  sint
C
The second equation models the charge q =
q(t) on a capacitor in an electrical circuit
containing a resistor and a capacitor, where
the current is driven by a sinusoidal
electromotive force sin ωt operating at
frequency ω.
Introduction
p

p'  rp 1  
K

In the third equation, called the logistic equation,
the state function p = p(t) represents the population
of an animal species in a closed ecosystem; r is the
population growth rate and K represents the
capacity of the ecosystem to support the
population.
Introduction
mx ' '   x
The fourth equation represents a model of
motion, where x = x(t) is the position of a
mass acted upon by a force −αx.
Introduction
• Differential Equation
– An equation containing the derivatives of one
or more dependent variables, with respect to
one or more independent variables, is said to
be a differential equation (DE).
• Classification
– By Type
– By Order
– By Linearity
Ordinary Differential Equations
• If an equation contains only ordinary
derivatives of one or more dependent
variables with respect to a single independent
variable, it is said to be an ordinary differential
equation (ODE).
2
d y dy

 6y  0
2
dx dx
dy
x
 5y  e
dx
dx dy

 2x  y
dt dt
Partial Differential Equations
• An equation involving partial derivatives of
one or more dependent variables of two or
more independent variables is called a partial
differential equation (PDE).
 2u  2u
 2 0
2
x
y
u
v

y
x
u  u
u
 2 2
2
x
t
t
2
2
Notation on ODEs
• Leibniz notation
• Prime notation
2
3
dy d y d y
, 2 , 3 , e tc.
dx dx dx
( 4)
y' , y' ' , y' ' ' , y , etc.
• Newton’s Dot Notation
2
ds
 32  s  32
2
dt
Notation on PDEs
• Leibniz Notation
 u  u
u
 2 2
2
x
t
t
• Subscript Notation
u xx  utt  2ut
2
2
Order of a Differential Equation
• The order of a differential equation (either
ODE or PDE) is the order of the highest
derivative in the equation.
3
d y  dy 
x

5

4
y

e


2
dx
 dx 
2
second
order
Order of a Differential Equation
• The order of a differential equation (either
ODE or PDE) is the order of the highest
derivative in the equation.
dy
4x
 xy
dx
first
order
First Order ODEs
• First-order ordinary differential equations are
occasionally written in the differential form
M(x, y)dx + N(x, y)dy = 0.
• For example:
dy
4x
 xy
dx
4xdy  ( x  y )dx
( x  y )dx  4xdy  0
( y  x)dx  4xdy  0
General and Normal Forms of ODEs
• In symbols, we can express an nth-order
ordinary differential equation in one
dependent variable by the general form
F(x, y, y' , y' ' , ..., y )  0
(n)
where F is a real-valued function of n+2
variables x, y, y’, …, y(n).
General and Normal Forms of ODEs
• In symbols, we can express an nth-order
ordinary differential equation in one
dependent variable by the normal form
n
d y
( n 1 )
 f ( x, y , y' , ..., y
)
n
dx
(n)
( n 1 )
 y  f ( x, y , y' , ..., y
)
where f is a real-valued continuous function.
General and Normal Forms of ODEs
• Example of ODE in general form:
y' ' y'6y  0
• Example of ODE in normal form:
y' '  y'6y
Linearity of ODEs
• An nth-order ordinary differential equation is
said to be linear if F is linear in y, y’, …, y(n):
dn y
d n 1 y
dy
a n ( x ) n  a n 1 ( x ) n 1  . . .  a1 ( x )
 a 0 ( x )y  g ( x )
dx
dx
dx
a n y ( n )  a n 1 y ( n 1 )  . . .  a1 y'  a 0 y  g( x )
Linearity of ODEs
• First-order ODE:
dy
a1 ( x )
 a 0 ( x )y  g ( x )
dx

a1 y'  a 0 y  g( x )
• Second-order ODE:
2
dy
dy
a 2 ( x) 2  a1 ( x)  a 0 ( x)y  g( x)
dx
dx

a 2 y' '  a 1 y'  a 0 y  g ( x )
Linearity of ODEs
n
n 1
d y
d y
dy
an ( x) n  an 1 ( x) n 1  . . .  a1 ( x)  a0 ( x)y  g( x)
dx
dx
dx
 an y n  an 1y n 1  . . .  a1y'a0 y  g( x)
• The dependent variable y and all its derivatives
y’, y’’, …, y(n), etc. are of the first degree (that is,
the power of each said term involving y is 1).
• The coefficients a0, a1, …, an of y, y’,…,yn
depend at most on the independent variable x.
Linearity of ODEs
• A linear, first order ODE:
(y  x)dx  4xdy  0
• A linear, second order ODE
y' '2y' y  0
• A linear, third order ODE
3
dy
dy
x
x
 5y  e
3
dx
dx
Nonlinear ODE
• A nonlinear ordinary differential equation
is simply one that is not linear.
• Nonlinear functions of the dependent
variable or its derivatives, such as sin(y)
or ey’, cannot appear in a linear
equation.
Nonlinear ODEs
• A nonlinear, first order ODE:
(1  y)y'2y  e
x
Nonlinear term:
coefficient depends
on y
• A nonlinear, second order ODE
2
dy
 siny  0
2
dx
Nonlinear term:
nonlinear function
of y
• A nonlinear, fourth order ODE
4
dy
2
y 0
4
dx
Nonlinear term:
power not 1
Nonlinear ODEs
A nonlinear ODE
 d4y

 dx4

3

 y0


Nonlinear term:
power not 1
Solution of an ODE
Any function , defined on an interval I and
possessing at least n derivatives that are
continuous on I, which when substituted
into an nth-order ordinary differential
equation reduces the equation to an
identity, is said to be a solution of the
equation on the interval.
Solution of an ODE
In other words, a solution of an nth-order
ordinary differential equation is a function 
that possesses at least n derivatives and
for which
F(x, (x), ' (x), ...,  (x))  0
(n )
for all x in I.
The Interval of the Solution
The Interval of the Solution is also called
– Interval of definition
– Interval of existence
– Interval of validity
– Domain of the solution
Solution Curve
• The graph of a solution  of an ordinary
differential equation is called a solution
curve.
• Since  is a differentiable function, it is
continuous on its interval I of definition.
Families of Solutions
• The study of differential equations is
similar to that of integral calculus.
• In some texts, a solution  is sometimes
referred to as an integral of the equation
and its graph is called an integral curve.
Families of Solutions
• When evaluating an anti-derivative or
indefinite integral in calculus, we use a
single constant c of integration.
• Analogously, when solving a first-order
differential equation F(x, y, y’) = 0, we
usually obtain a solution containing a
single arbitrary constant or parameter c.
Families of Solutions
• A solution containing an arbitrary constant
represents a set G(x, y, c) = 0 of solutions
called a one-parameter family of solutions.
• When solving an nth-order differential
equation F(x, y, y’, …, y(n)) = 0, we seek an
n-parameter family of solutions.
– We integrate n times, hence we get n arbitrary
constants.
Families of Solutions
• This means that a single differential
equation can possess an infinite number
of solutions corresponding to the unlimited
number of choices for the parameter(s).
• A solution of a differential equation that is
free of arbitrary parameters is called a
particular solution.
Families of Solutions
For example, the one-parameter family
y = cx – xcosx
is a solution of the linear first-order equation
xy’ – y = x2sinx
on the interval (-, ).
Families of Solutions
Check:
y = cx – xcosx
y’ = c – ( cosx + x(-sinx) )
y’ = c – cosx + xsinx
xy’ – y = x2sinx
x(c – cosx + xsinx) – (cx – xcosx) = x2sinx
cx – xcosx + x2sinx – cx + xcosx = x2sinx
x2sinx
= x2sinx
Families of Solutions
Some solutions of xy’ – y = x2sinx
Graph of y = cx – xcosx
Families of Solutions
If every solution of an nth-order ODE
F(x, y, y’, …, y(n)) on an interval I can be
obtained from an n-parameter family
G(x, y, c1, c2, …, cn) = 0 by appropriate
choices of the parameters ci, i = 1, 2, …,
n, we then say that the family is the
general solution of the differential
equation.
Singular Solution
• Sometimes a differential equation
possesses a solution that is not a member
of a family of solutions of the equation – that
is, a solution that cannot be obtained by
specializing any of the parameters in the
family of solutions.
• Such an extra solution is called a singular
solution.
Singular Solution
For example,
1 2

y   x  c
4

2
is a solution of the differential equation
dy
 xy
dx
on the interval (-, ).
1
2
Singular Solution
When c = 0, the resulting particular solution is
1 4
y x
16
But notice that the trivial solution y = 0 is a
singular solution, since it is not a member of
the family y = (1/2x2 + c)2; there is no way of
assigning a value to the constant c to obtain y
= 0.
Initial-Value Problems (IVPs)
We are often interested in problems in which
we seek a solution y(x) of a differential
equation so that y(x) satisfies prescribed
side conditions – that is, conditions imposed
on the unknown y(x) or its derivatives.
Initial-Value Problems (IVPs)
Solve:
n
d y
( n 1 )
 f ( x, y , y' , ...,y
)
n
dx
Subject to:
y(x 0 )  y 0
y' ( x 0 )  y 1

y ( n 1) ( x 0 )  y n 1
where y0, y1, …, yn-1 are arbitrarily specified
real constants.
Initial-Value Problems (IVPs)
• Such a problem is called an initial-value
problem (IVP).
• The values of y(x) and its first n-1 derivatives
at single point x0 (i.e., y(x0) = y0, y’(x0) = y1, …,
y(n-1)(x0) = yn-1) are called initial conditions.
First-Order IVP
Solve: dy
dx
Subject to:
 f ( x, y )
y( x 0 )  y 0
Second-Order IVP
Solve:
2
dy
 f ( x, y , y' )
2
dx
Subject to:
y( x 0 )  y 0
y' (x0 )  y 1
m = y1
Example
What function do you know from calculus
is such that its first derivative is itself?
Example
• Answer: y = Cex
• y = Cex
so that
y’ = Cex
and hence,
y = y’
• If we impose the
following initial
condition
y(0) = 3
then we have
y = Cex
3 = Ce0
3=C
Example
• Hence, y = 3ex is
a solution of the
IVP
y’ = y
y(0) = 3
• If we impose the
following initial
condition
y(1) = -2
then we have
y = Cex
-2 = Ce1
-2/e = C
Example
• Hence,
y = (-2/e)ex or
y = -2ex-1
is a solution of the
IVP
y’ = y
y(1) = -2
• The two solution
curves are shown
in dark blue and
dark red in the
next figure.
Example
The solution curves for
y  y'

y(0)  3
and
y  y'

y(1)  2
Existence of a Unique Solution
Theorem
Let R be a rectangular region in the xyplane defined by a  x  b, c  y  d that
contains the point (x0, y0) in its interior. If
f(x, y) and ∂f/∂y are continuous on R, then
there exist some interval I0: (xo - h, xo + h),
h > 0, contained in [a, b], and a unique
function y(x), defined on I0, that is a
solution of the initial-value problem.
Existence of a Unique Solution
Direction or Slope Fields
• Recall that a derivative dy/dx of a
differentiable function y = y(x) gives
slopes of tangent lines at points on its
graph.
• The derivative itself is also a function. In
other words,
dy
 f ( x, y )
dx
Direction or Slope Fields
• The function f in the normal form is called
the slope function or rate function.
• The value f(x,y) that the function f assigns
to a point (x,y) represents the slope of a
line; alternatively, we can envision it as a
line segment called a lineal element.
Direction or Slope Fields
• For example, say we’re given the
following function f and a point (2, 3) on
the solution curve:
 dy
  f ( x, y )  0.2xy
 dx

( 2,3)
• At the point (2, 3), the slope of a lineal
element is f(2,3) = 2(2)(3) = 1.2.
A lineal element at a point
A lineal element is tangent
to the solution curve that
passes through the point
in consideration
Direction or Slope Fields
If we systematically evaluate f over a
rectangular grid of points in the xy plane
and draw a line element at each point (x,y)
of the grid with slope f(x,y), then the
collection of all these lines is called a
direction field or a slope field of the
differential equation dy/dx = f(x,y).
Direction or Slope Fields
Visually, the direction field suggests the
appearance or shape of a family of
solution curves of the differential equation,
and consequently, it may be possible to
see at a glance certain qualitative aspects
of the solutions.
This figure shows a computer
generated direction field of dy/dx
= sin(x+y) over a region of the xy
plane. Note how the solution
curves shown in color follow the
flow of the field.
A single solution
curve that passes
through a direction
field must follow the
flow pattern of the
field. It is tangent to
a line element when
it intersects a point
in the grid.
Direction or Slope Fields
Going back to
dy
 f ( x, y )  0.2xy
dx
it can be shown that
y  ce
0.1x2
is a one-parameter family of solutions.
Direction Field
for dy/dx = 0.2xy
Some solution
curves in the
0.1x2
family y  ce
Solving Differential Equations
1. Analytical Approach
2. Qualitative Approach
3. Numerical Approach
Solving Differential Equations
1. Analytical
Approach
– Exact solution
using mathematical
principles
(calculus)
– Main focus of this
course (ENGIANA)
Solving Differential Equations
2. Qualitative Approach
–
Gleaning from the differential
equation answers to
questions such as
•
•
•
•
Does the DE actually have
solutions?
If a solution of the DE exists
and satisfies an initial condition,
is it the only such solution?
What are some of the properties
of the unknown solutions?
What can we say about the
geometry of the solution
curves?
Solving Differential Equations
3. Numerical Approach
– Use of numerical
methods and
computer algorithms
•
Can we somehow
approximate the
values of an unknown
solution?