Chapter 4 Higher Order Linear Equations 4.1 General Theory of the

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Chapter 4 Higher Order Linear Equations 4.1 General Theory of the
Chapter 4
Higher Order Linear Equations
4.1 General Theory of the nth Order Linear
Equations
Definition 1:
An nth order linear DE is an equation written in
the form
P0 (t ) y ( n )  P1 (t ) y ( n 1) 
Where
P0 (t ), P1 (t ),
, Pn (t )
 Pn (t ) y  G (t )
(1)
and G (t ) are continuous
real-valued functions on some interval I :   t 
 and P0 (t )
is nowhere zero in I . Then by dividing DE(1) by P0 (t )
y ( n )  p1 (t ) y ( n 1) 
 p n (t ) y  g (t )
1
(2)
Definition 2:
An initial value problem (IVP) consists of DE(2) with n
initial conditions
where
y (t 0 )  y 0 , y (t 0 )  y 0 , y (t 0 )  y 0,
, y ( n 1) (t 0 )  y 0( n 1 (3)
where t 0 is a point in I .
Theorem 4.1.1:
If the functions p1 (t ), p 2 (t ),
, p n (t )
and
g (t )
are
continuous real-valued functions on an open interval
I , then there exists exactly one solution y   (t ) of
the DE(2) that also satisfies the initial conditions (3).
This solution exists through the interval I .
Definition 3: A homogeneous equation is an
equation as in (2) where the term g (t )  0 i.e.
y ( n )  p1 (t ) y ( n 1) 
 p n (t ) y  0
2
(4)
Otherwise the DE is called a nonhomogeneous.
Theorem 4.1.2:
If the functions p1 , p 2 ,
, p n are continuous on
an open interval I , if the functions y 1 , y 2 ,
are solutions of DE(4) and if W ( y 1 , y 2 ,
,yn
,yn) 0
For at least one point in I then every solution of
DE can be expressed as a linear combination of
the solutions y 1 , y 2 ,
,yn .
Definition 4:
A set of solutions y 1 , y 2 ,
, y n of DE (4) where
the wronskian is nonzero is called a fundamental
set of solutions.
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The solution y  c1 y 1  c 2 y 2 
 c n y n is called
the general solution of the DE(4).
Definition 5:
The functions
f 1, f 2 ,
, f n are said to be linearly
dependent on I if there exist a set of constants
k 1, k 2 ,
, k n not all zeros, such that
k 1f 1  k 2f 2 
 k nf n  0 .
The functions
f 1, f 2 ,
, f n are said to be linearly
independent if they are not linearly dependent.
Remark:
y 1, y 2 ,
, y n is a fundamental set of solutions of
DE(4) if and only if they are linearly independent
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Example 1(Q5, Page 222): Determine intervals
where the solutions of the DEs exist.
(x  1) y (4)  (x  1) y   (tan x ) y  0.
Example 2(Q15, Page 222):Verify
that the give
functions are solutions of the DE and determine their
wronskian
xy   y   0,
1, x , x 3 .
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Theorem: Generalization of Abel's Theorem to higher order
Differential Equations.
If
y 1, y 2 ,
are solutions of the Differential
y ( n )  p1 (t ) y ( n 1) 
Equation
where
,yn
p1 , p 2 ,
 p n (t ) y  0
,
, p n are continuous on an open

interval I , then W ( y1 , y 2 ,..., y n )  ce
 p1 ( t ) dt
Example:
Use Abel's Theorem to find the wronskian of the
fundamental set of solutions of the differential equation
ty   2 y   y   ty  0
H.W. Problems 1-6, 11-16 Pages 222.
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4.2 Homogeneous Equations with Constant
Coefficients
Consider the nth order linear homogeneous DE
a0 y ( n )  a1 y ( n 1) 
Where
a0 , a1 ,
 an y  0
(1)
, an are constants.
y  e rt be a solution of DE(1) then
a0 r n  a1r n 1 
 an  0
(2)
is called the characteristic equation of the DE(1).
r ,r ,
Eq(2) has n roots 1 2
, rn .
1- Real and unequal roots.
The general solution
y  c1e r1t  c 2e r2t 
7
 c ne rn t
(3)
Example 1: Find the general solution of the DE
y (4)  y   7 y   y   6 y  0, y (0)  1,
y (0)  0, y (0)  2, y (0)  1
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Complex Roots:
If the characteristic equation has complex roots,
they must occur in conjugate pairs    i
Example 2: Find the general solution of the DE
y (4)  y  0.
Repeated Roots:
If the characteristic equation has s repeated
roots, e rt , te rt , t 2e rt , , t s 1e rt .
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If the complex root    i is repeated s times,
then its conjugate    i is repeated s times.
Example 3: Find the general solution of the DE
y (4)  2 y   y  0.
Example 4: Find the general solution of the DE
y (4)  y  0.
H.W. Problems 7-37 Pages 230.
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1
3
Example 5 (Q.7, Page 230): Find the roots of 1
Example 6: Find the general solution of the DE
y   5 y   3 y   y  0.
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4.3 The Method of Undetermined
Coefficients
This method as we have discussed in Section
3.6 and the main difference that we use it for
higher order DE.
Example 1: Find the general solution of the DE
y   3y   3y   y  4e t .
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Example 2: Find a particular solution of the DE
y (4)  2 y   y  3sin t  5cos t .
Do not find the constants.
Example 3: Find a particular solution of the DE
y   4 y   t  3cos t  e 2t .
Do not find the constants.
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Example 4: Without finding the constants, find a
formula for a particular solution of the
nonhomogeneous DE if
(a)
y c  c1  c 2e t  c 3te t  c 4 sin 2t  c 5 cos 2t
(i)
g (t )  3e t  2t cos t
(ii)
g (t )  3e t  2t cos t
(b)
y c  c1  c 2t  c 3t 2  (c 4  c 5t ) sin 3t  (c 6  c 7t ) cos 3t
g (t )  (t 2  1)  t sin 3t
H.W. Problems 1-17 Pages 235.
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Example 5 (Q.5, Page 235): Determine the general
solution of the DE
y (4)  4 y   t 2  e t .
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4.4 The Method of Variation of
Parameters
This method is a generalization of the method for
second order DE from Section 3.7.
Consider the DE
y ( n )  p1 (t ) y ( n 1) 
 p n (t ) y  g (t )
(1)
To find a particular solution, at first we solve the
corresponding homogeneous DE
y ( n )  p1 (t ) y ( n 1) 
 p n (t ) y  0
(2)
Let
y c  c1 y 1  c 2 y 2 
 cn y n
Be the general solution of the DE(2)
Then
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(3)
y p  u1 y 1  u 2 y 2 
un y n
(4)
Where
u1 , u 2 ,
,u n are functions of t and
W m (t ) g (t )
u m (t )  
dt , m  1, 2,
W (t )
,n
where
W (t ) W ( y 1 , y 2 ,
, y n )(t )
W m (t ) is the determinate obtained from W (t ) by
replacing the mth column by the column (0, 0,
17
,1) .
Example 1(Q1, Page 240): Find the general solution of
the DE
y   y   tan t , 0  t   .
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Example 2: Solve the DE
y   y   t .
H.W. Problems 1-17 Pages 240.
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