72 72 Chapter 3 Second Order Linear Differential Equations Sec 3.1 : Homogeneous equation A linear second order differential equations is written as When d(x) = 0, the equation is called homogeneous, otherwise it is called nonhomogeneous. , For the study of these equations sometimes we will consider the standard forms given by ) ( ) ( ' ) ( ' ' x g y x q y x p y 0 ) ( ' ) ( ' ' y x q y x p y Linear differential operators(ways to write the linear 2 nd ODE): In mathematics a function that ``transforms a function into a different function'' is called an operator . (1) L -Operator Let y'' + p(x)y' + q(x)y = g(x) be a second order linear differential equation. Then we call the operator L(y) = y'' + p(x)y' + q(x)y the corresponding linear operator (Differential operator). Thus in this chapter we want to find solutions to the equation L(y) = g(x) y(x 0 ) = y 0 y'(x 0 ) = y' 0
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Chapter 3
Second Order Linear Differential Equations
Sec 3.1 : Homogeneous equation
A linear second order differential equations is written as
When d(x) = 0, the equation is called homogeneous, otherwise it is called nonhomogeneous.
,
For the study of these equations sometimes we will consider the standard forms given by
)()(')('' xgyxqyxpy
0)(')('' yxqyxpy
Linear differential operators(ways to write the linear 2nd ODE):
In mathematics a function that ``transforms a function into a different function'' is called an operator.
(1) L -Operator
Let y'' + p(x)y' + q(x)y = g(x)
be a second order linear differential equation. Then we call the operator
L(y) = y'' + p(x)y' + q(x)y
the corresponding linear operator(Differential operator). Thus in this chapter we want to find solutions to
the equation
L(y) = g(x) y(x0) = y0 y'(x0) = y'0
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Proof:
(1)
(2) the proof of (2) is exercise #28 sec 4.2
Note: Any operator that satisfies (1) & (2) is called linear operator. If (1) & (2) fails to hold, the operator is
nonlinear
(2) D- Operator
Recall from calculus that
dx
dyDy ,
2
22
dy
ydyD , and in general
n
nn
dy
ydyD . Using this notations, we can express L as
])[(][ 22 yqpDDqypDyyDyL
for example, if
,3'4''][ yyyyL
then we can write
Linearity of the differential operator L :
Let L(y) = y'' + p(x)y' + q(x)y. if y , 1y and 2y are any twice-differeniable functions on the interval I
and if c is any constant, then
(1) )()()( 2121 yLyLyyL ,
(2) )()( ycLcyL
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])[34(34][ 22yDDyDyyDyL
when p and q are constants we can treat qypDyyD 2 as polynomial , so for the above example
we can write
yDDyDDyDyyDyL )3)(1(])[34(34][ 22
Exercises:
# 3.1.1 Let yyyL ''][2 compute
(a) ][sin2 xL (b) ][2
2 xL (c) ][2r
xL (d) ]2[2
2x
eL
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Sec 3.2: Solution of Homogeneous Linear Second Order Differential
Equations
Theory of Solutions
Here we will investigate solutions to homogeneous differential equations. Consider the homogeneous linear
differential equation
We have the following results:
(1)
The above theorem makes clear the importance of being able to
decide whether or not a set of solutions )(1 xy and )(2 xy of
Are linearly independent.
Methods to check if two functions are Linearly independent:
(1) Linear Dependence and Indepedence using Definition :
Definition:(Linear dependence)
Two functions 22 yandy are said to be linearly independent on the interval I, if
there exist nonzero constants 22 candc such that for all x in I
02211 ycyc
General Solution for a Second-Order Homogeneous Linear Equation: Theorem: The 2nd order homogeneous equation
0)(')('')( yxcyxbyxa (1)
always has exactly two linearly independent solutions )(1 xy and )(2 xy . The general
solution to equation (1) is given by the formula
2211)( ycycxy
where the functions 21, yy is the pair of linearly independent solutions to
equation (1).
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f 22 yandy are linearly
independent then 021 cc is the only solution for
02211 ycyc
Another way of stating this definition is ,
ratio test as follows:
Two functions 22 yandy are linearly dependent in I if cy
y
2
1
Two functions 22 yandy are linearly independent in I if cy
y
2
1
Remark: Proportionality of two functions is equivalent to their linear dependence.
(2) Linear Dependence and Independence using Wronskian:
It is not easy to use the above definition for higher order, Instead we can use the wronskian.
The Wronskian : for two functions
Let and be two differentiable functions. and are linearly dependent(proportional) if and only if
cy
y
2
1 . an equivalent criteria for linearly dependent can be derived from the followingt:
and are linearly dependent
So cy
y
2
1
Differentiate both sides 02
1
y
y
.
is called the Wronskian of of and , that is
Wronskian of and is
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Therefore, we have the following:
Now we have the following conclusion:
Let and be two differentiable functions. The Wronskian , associated to and , is the
function
We have the following important properties:
(1) If and are two solutions of the equation y'' + p(x)y' + q(x)y = 0, then
In this case, we say that and are linearly independent.
(2) If and are two linearly independent solutions of the equation y'' + p(x)y' + q(x)y = 0, then
any solution y is given by
for some constant and . In this case, the set is called the fundamental set of solutions.
Exercises:
# 3.1.3 Determine whether the given solutions of the following DE are linearly independent
(a) xyxyyy 3cos3sin09'' 21
(b) xx
eyeyyy 20'' 21
(c) xx
eyeyyyy2
21 302''
There is a fascinating relationship between second order linear differential equations and the
Wronskian. This relationship is stated below.
Two functions 1y and 2y are linearly dependent if 0),( 21 yyW
0),( 21 yyWif dependentinlinearly are 2yand 1yTwo functions
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Theorem: Abel's Theorem
Let y1 and y2 be solutions on the differential equation
L(y) = y'' + p(x)y' + q(x)y = 0
where p and q are continuous on [a,b]. Then the Wronskian is given by
dxxp
CexyyW)(
21 ))(,(
where C is a constant depending on only y1 and y2, but not on x. The Wronskian is
either zero for all x in [a,b] or always positive, or always negative.