Transcript Math 240: Transition to Advanced Math

Math 3120
Differential Equations
with
Boundary Value Problems
Chapter 1
Introduction to Differential
Equations
Basic Mathematical Models
Many physical systems describing the real world are
statements or relations involving rate of change. In
mathematical terms, statements are equations and rates
are derivatives.
Definition: An equation containing derivatives is called a
differential equation.
Differential equation (DE) play a prominent role in
physics, engineering, chemistry, biology and other
disciplines. For example: Motion of fluids, Flow of
current in electrical circuits, Dissipation of heat in solid
objects, Seismic waves, Population dynamics etc.
Definition: A differential equation that describes a physical
process is often called a mathematical model.
Basic Mathematical Models

Formulate a mathematical model describing motion
of an object falling in the atmosphere near sea level.

Variables: time t, velocity v

dv
Newton’s 2nd Law: F = ma =m dt

Force of gravity: F = mg
force

Force of air dv
resistance: F =  v
m  mg   v
force
dt
Then

net force
downward
upward
Basic Mathematical Models

We can also write Newton’s 2nd Law:
ds 2
F m 2
dt


whe re s 
dv
dt
where s(t) is the distance the body falls in time t
from its initial point of release
Then,
d 2s
ds
m 2 
 mg
dt
dt
Examples of DE
dv
m  mg
(1)   v
dt
d 2s
ds
m 2 
 mg
dt
dt
d 2q
ds 1
L 2  R  q  E (t )
dt
dt C
 2 u ( x, t )  2 u ( x, t )


(heat equation)
2
x
t
2
2

u
(
x
,
t
)

u ( x, t )
a2

(wave equation)
2
2
x
t
(2)
(3)
2
(4)
(5)
Classifications of Differential Equation

By Types
 Ordinary
 Partial
Differential Equation (ODE)
Differential Equation (PDE)

Order

Systems

Linearity
 Linear
 Non-Linear
Ordinary Differential Equations

When the unknown function depends on a single
independent variable, only ordinary derivatives appear
in the equation. In this case the equation is said to be
an ordinary differential equations.

For example:

A DE can contain more than one dependent variable.
For example:
dv
 9.8  0.2v,
dt
d 2 y dy
  0.5 y  0
2
dx
dx
dx dy

 x y
dt dt
Partial Differential Equations

When the unknown function depends on several
independent variables, partial derivatives appear in
the equation. In this case the equation is said to be a
partial differential equation.

2
2
Examples:

u
(
x
,
t
)

u ( x, t )
2


(heat equation)
2
x
t
2
2

u
(
x
,
t
)

u ( x, t )
2
a

(wave equation)
2
2
x
t
Notation

Leibniz
dy d 2 y d 3 y
dny
, 2 , 3 ,........ n
dx dx dx
dx
y, y, y, y , y

Prime

Dot

Subscript u ,
x
( 4)
dy
 y ,
dx
u xx ,
d2y
 y
2
dx
u yy
( n 1)
,y
(n)
Systems of Differential Equations

Another classification of differential equations depends on
the number of unknown functions that are involved.

If there is a single unknown function to be found, then one
equation is sufficient. If there are two or more unknown
functions, then a system of equations is required.

For example, Lotka-Volterra (predator-prey) equations
have the form
du / dt  a u   uv
dv / dt  cv   uv
where u(t) and v(t) are the respective populations of prey
and predator species. The constants a, c, ,  depend on
the particular species being studied.
Order of Differential Equations

The order of a differential equation is the order of
the highest derivative that appears in the equation.

Examples:
y  3 y  0

d4y d2y
2t


1

e
dt 4 dt 2
An nth order differential equation can be written as


F t , y, y, y, y,, y ( n )  0

u xx  u yy  sin t
The( n )normal form of Eq. (6) is( n1)

y (t )  f t , y, y , y , y ,, y
(6)
(7)
Linear & Nonlinear Differential Equations

An ordinary differential equation


F t , y, y, y, y,, y ( n)  0
is linear if F is linear in the variables
y, y, y, y,, y ( n )

Thus the general linear ODE has the form
a0 (t ) y ( n )  a1 (t ) y ( n 1)    an (t ) y  g (t )

The characteristic of linear ODE is given as
Linear & Nonlinear Differential Equations

Example: Determine whether the equations below
are linear or nonlinear.
(1) y   3 y  0
d4y
d2y
(4) 4  t 2  1  t 2
dt
dt
(2) y   3e y y   2t  0
(5) u xx  uu yy  sin t
(3) y   3 y   2t 2  0
(6) u xx  sin( u )u yy  cos t
Solutions to Differential Equations

A solution of an ordinary differential equation

y ( n ) (t )  f t , y, y , y , y ,, y ( n1)

(7)
 ,  ,,  ( n1) ,  ( n)
on an interval I is a function (t) such that
exists and satisfies the equation:
 ( n) (t )  f t ,  ,  ,  ,,  ( n1) 
for every t in I.

Unless stated we shall assume that function f of Eq. (7) is a
real valued function
and
y 
(t ) we are interested in obtaining real
valued solutions

NOTE: Solutions of ODE are always defined on an interval.
Solutions to Differential Equations

y (t )  cos t
Example: Show that
is a solution of the
ODE y   y  0 on the interval (-∞, ∞).
y (t )  sin t

Verify that
on the interval (-∞, ∞).
y   y  0
is a solutions of the ODE
Types of Solutions

Trivial solution: is a solution of a differential equation
that is identically zero on an interval I.

Explicit solution: is a solution in which the dependent
variable is expressed solely in terms of the
independent variable and constants. For example,
y (t )  cos t , and
y (t )  sin t
y   y  0
are two explicit solutions of the ODE

Implicit solution is a solution that is not in explicit
form.
Families of Solutions

F x, y, y  0
A solution of a first- order differential equation
usually contains a single arbitrary constant or
parameter c.


Gx, y, c  0
One-parameter family of solution: is a solution
containing an arbitrary constant represented by a set
of solutions.
Particular solution: is a solution of a differential
equation that is free of arbitrary parameters.
Initial Value Problems (IVP)

Initial Conditions (IC) are values of the solution and /or its
derivatives at specific points on the given interval I.

A differential equation along with an appropriate number of
IC is called an initial value problem. Generally, a first order
differential equation is of the typey'  f (t , y),
y(t )  y

An nth order IVP is of the form
0
y ( n )  f (t , y, y' ,....., y ( n1) )
subject to
where

y(t 0 )  y 0 ,
y 0 , y1 ,...., y n1
y' (t 0 )  y1 ,...., y ( n1) (t 0 )  y n1
are arbitrary constants.
Note: The number of IC’s depend on the order of the DE.
0
Solutions to Differential Equations

Three important questions in the study of
differential equations:

Is there a solution? (Existence)

If there is a solution, is it unique? (Uniqueness)

If there is a solution, how do we find it?
(Qualitative Solution, Analytical Solution, Numerical
Approximation)
Theorem 1.2.1: Existence of a Unique
Solution

Suppose f and f/y are continuous on some open
rectangle R defined by (t, y)  (,  ) x (,  ) containing
the point (t0, y0). Then in some interval (t0 - h, t0 + h) 
(,  ) there exists a unique solution y = (t) that
satisfies the IVP
y '  f (t , y )
subject to y (t 0 )  y 0

It turns out that conditions stated in Theorem 1.2.1 are
sufficient but not necessary.