Guide to Differential Equations Admission 2015

Admission in India 2015
By:
admission.edhole.com

Ordinary Differential
Equations
S.-Y. Leu
Sept. 21,28, 2005

1.1 Definitions and Terminology
1.2 Initial-Value Problems
1.3 Differential Equation as
Mathematical Models
CHAPTER 1
Introduction 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(.
)Zill, Definition 1.1, page 6(.
1.1Definitions and
Terminology

Recall Calculus
Definition of a Derivative
If , the derivative of or
With respect to is defined as
The derivative is also denoted by or
1.1Definitions and
Terminology
)(xfy = y )(xf
x
h
xfhxf
dx
dy
h
)()(
lim
0
−+
=
→
dx
df
y ,'
)('
xf

Recall the Exponential function
dependent variable: y
independent variable: x
1.1Definitions and
Terminology
x
exfy 2
)( ==
ye
dx
xd
e
dx
ed
dx
dy xx
x
22
)2()( 22
2
==





==

Differential Equation:
Equations that involve dependent variables and
their derivatives with respect to the independent
variables.
Differential Equations are classified
by
type, order and linearity.
1.1Definitions and
Terminology

Differential Equations are classified by
type, order and linearity.
TYPE
There are two main types of differential
equation: “ordinary” and “partial”.
1.1Definitions 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
1.1Definitions and
Terminology
x
ey
dx
dy
=+ 5 062
2
=+− y
dx
dy
dx
yd yx
dt
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
1.1Definitions and
Terminology
t
u
t
u
x
u
∂
∂
−
∂
∂
=
∂
∂
22
2
2
2
x
v
y
u
∂
∂
−=
∂
∂

ORDER
The order of a differential equation is the
order of the highest derivative found in
the DE.
second order first order
1.1Definitions and
Terminology
x
ey
dx
dy
dx
yd
=−





+ 45
3
2
2

1.1Definitions and
Terminology
x
eyxy =− 2'
3''
xy =
0),,( '
=yyxFfirst 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.
1.1Definitions and
Terminology
)1('
,..., −n
yyy
)()()(...)()( 011
1
1 xgyxa
dx
dy
xa
dx
yd
xa
dx
yd
xa n
n
nn
n
n =++++ −
−
−
0),......,,( )('
=n
yyyxF

LINEAR or NONLINEAR
or
linear first-order ordinary differential equation
linear second-order ordinary differential
equation
 linear third-order ordinary differential equation
1.1Definitions and
Terminology
0)(4 =−+ xy
dx
dy
x
02 '''
=+− yyy
04)( =+− xdydxxy
x
ey
dx
dy
x
dx
yd
=−+ 533
3

LINEAR or NONLINEAR
coefficient depends on y
nonlinear first-order ordinary differential equation
nonlinear function of y
nonlinear second-order ordinary differential equation
power not 1
 nonlinear fourth-order ordinary differential equation
1.1Definitions and
Terminology
0)sin(2
2
=+ y
dx
yd
x
eyyy =+− 2)1( '
02
4
4
=+ y
dx
yd

LINEAR or NONLINEAR
NOTE:
1.1Definitions and
Terminology
...
!7!5!3
)sin(
753
+−+−=
yyy
yy ∞<<−∞ x
...
!6!4!2
1)cos(
642
+−+−=
yyy
y ∞<<−∞ x

Solutions of ODEs
DEFINITION: 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 n-th order ODE
reduces the equation to an identity, is said to be a
solution of the equation on the interval.
)Zill, Definition 1.1, page 8(.
1.1Definitions and
Terminology
φ

Namely, a solution of an n-th order ODE is a
function which possesses at least n
derivatives and for which
for all x in I
We say that satisfies the differential equation
on I.
1.1Definitions and
Terminology
0))(),(),(,( )('
=xxxxF n
φφφ

Verification of a solution by substitution
Example:

left hand side:
right-hand side: 0
The DE possesses the constant y=0 trivial solution
1.1Definitions and
Terminology
xxxx
exeyexey 2, '''
+=+=
x
xeyyyy ==+− ;02 '''
0)(2)2(2 '''
=++−+=+− xxxxx
xeexeexeyyy

DEFINITION: solution curve
A graph of the solution of an ODE is called a
solution curve, or an integral curve of
the equation.
1.1Definitions and
Terminology

1.1Definitions and Terminology
DEFINITION: families of solutions
A solution containing an arbitrary constant
(parameter) represents a set of
solutions 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
0),,( =cyxG
0),......,,( )('
=n
yyyxF

Example:
1.1Definitions and
Terminology
2'
=+ yy
x
kex −
+= 2)(φ
2'
=+ yy
x
kex −
+= 2)(φ
x
kex −
−=)(φ'
22)(φ)(φ'
=++−=+ −− xx
kekexx


Figure 1.1 Integral curves of y’+ y = 2 for k = 0, 3, –3, 6, and –6.
©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
x
key −
+= 2

Example:
1.1Definitions and
Terminology

1'
+=
x
y
y
Cxxxx += )ln()(φ 0>x
Cxx ++= 1)ln()(φ'
1
)(φ
1
)ln(
)(φ'
+=+
+
=
x
x
x
Cxxx
x
for all, 

Figure 1.2 Integral curves of y’ + ¹ y =
ex
for c =0,5,20, -6, and –10. x
)(
1
cexe
x
y xx
+−=

Second-Order Differential Equation
Example:
is a solution of
By substitution:
)4sin(17)4cos(6)(φ xxx −=
016''
=+ xy
0φ16φ
)4sin(272)4cos(96φ
)4cos(68)4sin(24φ
''
''
'
=+
+−=
−−=
xx
xx
0),,,( '''
=yyyxF
( ) 0)(φ),(φ),(φ, '''
=xxxxF

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(

IF only try x=x1, andx=x2

xy 12''
=
KCxxy ++= 3
2
KCxxxy
KCxxxy
++=
++=
2
3
22
1
3
11
2)(
2)(

Figure 2.1 Graphs of y = 2x³ + C x +K for various values of C and K.
e.g. X=0, y=k

Figure 2.2 Graphs of y = 2x³ + C x + 3 for various values of C.
To satisfy the I.C. y(0)=3
The solution curve must
pass through (0,3)
Many solution curves through (0,3)

Figure 2.3 Graph of y = 2x³ - x + 3.
To satisfy the I.C. y(0)=3,
y’(0)=-1, the solution curve
must pass through (0,3)
having slope -1

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 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.
1.1Definitions and
Terminology

DEFINITION: implicit solution
A relation is said to be an
implicit solution of an ODE on an interval I
provided there exists at least one function
that satisfies the relation as well as the
differential equation on I.
a relation or expression that
defines a solution implicitly.
In contrast to an explicit solution
1.1Definitions and Terminology
0),( =yxG
)(xy φ=
φ
φ
0),( =yxG

Verify by implicit differentiation that the given
equation implicitly defines a solution of the
differential equation
1.1Definitions and
Terminology
Cyxxxyy =−−−+ 232 22
0)22(34 '
=−++−− yyxxy

Verify by implicit differentiation that the given
equation implicitly defines a solution of the
1.1Definitions and
Terminology
Cyxxxyy =−−−+ 232 22
0)22(34 '
=−++−− yyxxy
0)22(34
02234
02342
/)(/)232(
'
'''
'''
22
=−++−−==>
=−++−−==>
=−−−++==>
=−−−+
yyxxy
yyyxyxy
yxxyyyy
dxCddxyxxxyyd

Conditions
Initial Condition: Constrains that are specified at the
initial point, generally time point, are called initial
conditions. Problems with specified initial conditions
are called initial value problems.
Boundary Condition: Constrains that are specified
at the boundary points, generally space points, are
called boundary conditions. Problems with specified
boundary conditions are called boundary value
problems.
1.1Definitions and
Terminology

First- and Second-Order IVPS
Solve:
Subject to:
Solve:
Subject to:
1.2Initial-Value Problem
00 )( yxy =
),( yxf
dx
dy
=
),,( '
2
2
yyxf
dx
yd
=
10
'
00 )(,)( yxyyxy ==

DEFINITION: initial value problem
An initial value problem or IVP is a
problem which consists of an n-th order
ordinary differential equation along with n
initial conditions defined at a point found
in the interval of definition
initial conditions
where are known constants.
0x
),...,,,( )1(' −
= n
n
n
yyyxf
dx
yd
I
10
)1(
10
'
00 )(,...,)(,)( −
−
=== n
n
yxyyxyyxy
110 ,...,, −nyyy

THEOREM: Existence of a Unique
Solution
Let R be a rectangular region in the xy-plane
defined by that contains
the point in its interior. If and
are continuous on R, Then there
exists some interval
contained in and
a unique function defined on
that is a solution of the initial value problem.
dycbxa ≤≤≤≤ ,
),( 00 yx ),( yxf
yf ∂∂ /
0,: 000 >+<<− hhxxhxI
bxa ≤≤
)(xy 0I

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