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Math 3120 Differential Equations
withBoundary Value
Problems
Chapter 1Introduction to Differential
Equations
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Basic Mathematical ModelsMany 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.
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Formulate a mathematical model describing motion of an object falling in the atmosphere near sea level.
Variables: time t, velocity v Newton’s 2nd Law: F = ma = net
force
Force of gravity: F = mg downward force
Force of air resistance: F = v upward force
Then
vmgdt
dvm
Basic Mathematical Models
dt
dvm
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We can also write Newton’s 2nd Law:
where s(t) is the distance the body falls in time t from its initial point of release
Then,
Basic Mathematical Models
dt
dvs
dt
dsmF re whe
2
2
mgdt
ds
dt
sdm
2
2
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(1)
(2)
(3)
(4)
(5)
mgdt
ds
dt
sdm
2
2
Examples of DE
vmgdt
dvm
equation) (wave ),(),(
equation)(heat ),(),(
2
2
2
22
2
2
22
t
txu
x
txua
t
txu
x
txu
)(1
2
2
tEqCdt
dsR
dt
qdL
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Classifications of Differential Equation
By Types Ordinary Differential Equation (ODE) Partial Differential Equation (PDE)
Order Systems Linearity
Linear Non-Linear
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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:
05.0,2.08.92
2
ydx
dy
dx
ydv
dt
dv
yxdt
dy
dt
dx
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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.
Examples:
equation) (wave ),(),(
equation)(heat ),(),(
2
2
2
22
2
2
22
t
txu
x
txua
t
txu
x
txu
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Notation
Leibniz
Prime
Dot
Subscript
)()1()4( ,,,,, nn yyyyyy
n
n
dx
yd
dx
yd
dx
yd
dx
dy,........,,
3
3
2
2
ydx
ydy
dx
dy 2
2
,
yyxxx uuu ,,
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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
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.
uvcvdtdv
uvuadtdu
/
/
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Order of Differential Equations
The order of a differential equation is the order of the highest derivative that appears in the equation.
Examples:
An nth order differential equation can be written as
The normal form of Eq. (6) is
tuuedt
yd
dt
ydyy yyxx
t sin 1 03 22
2
4
4
)7( ,,,,,,)( )1()( nn yyyyytfty
(6) 0,,,,,, )( nyyyyytF
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Linear & Nonlinear Differential Equations
An ordinary differential equation
is linear if F is linear in the variables
Thus the general linear ODE has the form
The characteristic of linear ODE is given as
0,,,,,, )( nyyyyytF
)(,,,,, nyyyyy
)()()()( )1(1
)(0 tgytaytayta n
nn
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Linear & Nonlinear Differential Equations
Example: Determine whether the equations below are linear or nonlinear.
tuuutyy
tuuutyey
tdt
ydt
dt
ydyy
yyxx
yyxxy
cos)sin()6(023)3(
sin)5(023)2(
1)4(03)1(
2
22
2
4
4
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Solutions to Differential Equations
A solution of an ordinary differential equation
on an interval I is a function (t) such that
exists and satisfies the equation:
for every t in I.
Unless stated we shall assume that function f of Eq. (7) is a real valued function and we are interested in obtaining real valued solutions
NOTE: Solutions of ODE are always defined on an interval.
)1()( ,,,,,)( nn tft
)()1( ,,,, nn )7( ,,,,,,)( )1()( nn yyyyytfty
)(ty
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Solutions to Differential Equations
Example: Show that is a solution of the ODE on the interval (-∞, ∞).
Verify that is a solutions of the ODE on the interval (-∞, ∞).
tty sin)(
tty cos)( 0 yy
0 yy
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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,
are two explicit solutions of the ODE
Implicit solution is a solution that is not in explicit form.
ttytty sin)( and ,cos)( 0 yy
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Families of Solutions
A solution of a first- order differential equation
usually contains a single arbitrary constant or parameter c.
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.
0,, yyxF
0,, cyxG
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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 type
An nth order IVP is of the form
where are arbitrary constants. Note: The number of IC’s depend on the order of the DE.
10)1(
1000
)1()(
)(,....,)(',)( subject to
),.....,',,(
nn
nn
ytyytyyty
yyytfy
00 )( ),,(' ytyytfy
110 ,....,, nyyy
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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)
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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
It turns out that conditions stated in Theorem 1.2.1 are sufficient but not necessary.
00 )( subject to
),('
yty
ytfy