ordinary differential equations [email protected]
TRANSCRIPT
1.1 Definitions and Terminology
1.2 Initial-Value Problems 1.3 Differential Equation as Mathematical Models
CHAPTER 1Introduction to Differential Equations
DEFINITION: differential equation
An equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a differential equation (DE) .
(Definition 1.1, page 6.)
1.1 Definitions and Terminology
Recall Calculus
Definition of a Derivative
If , the derivative of orWith respect to is defined as
The derivative is also denoted by or
1.1 Definitions and Terminology
)(xfy y )(xfx
h
xfhxf
dx
dyh
)()(lim
0
dx
dfy ,' )(' xf
Recall the Exponential function
dependent variable: y independent variable: x
1.1 Definitions and Terminology
xexfy 2)(
yedx
xde
dx
ed
dx
dy xxx
22)2()( 22
2
Differential Equation: Equations that involve dependent
variables and their derivatives with respect to the
independentvariables.
Differential Equations are classified by
type, order and linearity.
1.1 Definitions and Terminology
Differential Equations are classified by
type, order and linearity.
TYPEThere are two main types of differential equation: “ordinary”
and “partial” .
1.1 Definitions and Terminology
Ordinary differential equation (ODE)
Differential equations that involve only ONE independent variable are called ordinary differential equations.
Examples:
, , and
only ordinary (or total ) derivatives
1.1 Definitions and Terminology
xeydx
dy5 06
2
2
ydx
dy
dx
yd yxdt
dy
dt
dx 2
Partial differential equation (PDE)
Differential equations that involve two or more independent variables are
called partial differential equations.Examples:
and
only partial derivatives
1.1 Definitions and Terminology
t
u
t
u
x
u
22
2
2
2
x
v
y
u
ORDERThe order of a differential equation is the order of the highest derivative found in the DE.
second order first order
1.1 Definitions and Terminology
xeydx
dy
dx
yd
45
3
2
2
1.1 Definitions and Terminology
xeyxy 2'
3'' xy
0),,( ' yyxFfirst order
second order
0),,,( ''' yyyxF
Written in differential form: 0),(),( dyyxNdxyxM
LINEAR or NONLINEAR
An n-th order differential equation is said to be linear if the function
is linear in the variables
there are no multiplications among dependent variables and their derivatives. All coefficients are functions of independent variables.
A nonlinear ODE is one that is not linear, i.e. does not have the above form.
1.1 Definitions and Terminology
)1(' ,..., nyyy
)()()(...)()( 011
1
1 xgyxadx
dyxa
dx
ydxa
dx
ydxa
n
n
nn
n
n
0),......,,( )(' nyyyxF
LINEAR or NONLINEAR
orlinear first-order ordinary differential equation
linear second-order ordinary differential equation
linear third-order ordinary differential equation
1.1 Definitions and Terminology
0)(4 xydx
dyx
02 ''' yyy
04)( xdydxxy
xeydx
dyx
dx
yd 53
3
3
LINEAR or NONLINEAR
coefficient depends on ynonlinear first-order ordinary differential equation
nonlinear function of ynonlinear second-order ordinary differential equation
power not 1 nonlinear fourth-order ordinary differential equation
1.1 Definitions and Terminology
0)sin(2
2
ydx
yd
xeyyy 2)1( '
024
4
ydx
yd
LINEAR or NONLINEAR
NOTE:
1.1 Definitions and Terminology
...!7!5!3
)sin(753
yyy
yy x
...!6!4!2
1)cos(642
yyy
y x
Verification of a solution by substitution Example:
left hand side:
right-hand side: 0
The DE possesses the constant y=0 trivial solution
1.1 Definitions and Terminology
xxxx exeyexey 2, '''
xxeyyyy ;02 '''
0)(2)2(2 ''' xxxxx xeexeexeyyy
1.1 Definitions and Terminology
DEFINITION: families of solutions A solution containing an arbitrary constant (parameter) represents a set ofsolutions to an ODE called a one-parameter family of solutions .
A solution to an n−th order ODE is a n-parameter family of solutions
.
Since the parameter can be assigned an infinite number of values, an ODE can have an infinite number of solutions.
0),,( cyxG
0),......,,( )(' nyyyxF
Verification of a solution by substitution Example:
1.1 Definitions and Terminology
2' yyxkex 2)(φ
2' yyxkex 2)(φxkex )(φ '
22)(φ)(φ ' xx kekexx
Second-Order Differential Equation
Consider the simple, linear second-order equation
,
To determine C and K, we need two initial conditions, one specify a point lying on the solution curve
and the other its slope at that point, e.g, .
WHY???
012'' xy
xy 12'' Cxxdxdxxyy 2'' 612)('
KCxxdxCxdxxyy 32' 2)6()(
Cy )0('Ky )0(
Second-Order Differential Equation
IF only try x=x1, and x=x2
It cannot determine C and K,
xy 12'' KCxxy 32
KCxxxy
KCxxxy
23
22
13
11
2)(
2)(
Solutions General Solution: Solutions obtained from integrating the differential equations are called general solutions. The general solution of a nth order ordinary differential equation contains n arbitrary constants resulting from integrating n
times. Particular Solution: Particular solutions are the solutions obtained by assigning specific values to
the arbitrary constants in the general solutions. Singular Solutions: Solutions that can not be expressed by the general solutions are called
singular solutions.
1.1 Definitions and Terminology