ODE Lecture Notes
Section 4.1
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Section 4.1: General Theory of nth Order Linear Equations
Big Idea: The techniques and theorems regarding second-order linear differential equations can
be extended to higher-order differential equations.
Big Skill: You should be able to verify solutions, compute Wronskians, and determine linear
independence of solutions.
Usual form of an nth order linear differential equation:
dny
d n 1 y
dy
P0  t  n  P1  t  n 1   Pn 1  t   Pn  t  y  G  t 
dt
dt
dt
Assumptions:
 The functions P0  t  ,
, Pn  t  , G  t  are continuous and real-valued on some interval
I :   t   , and P0  t   0 for any t  I .
Linear differential operator form:
dny
d n 1 y
L  y   n  p1  t  n1 
dt
dt
 pn 1  t 
dy
 pn  t  y  g  t 
dt
Notes:
 Solving an nth order equation ostensibly requires n integrations
 This implies n constants of integration
 Also implies n initial conditions to completely specify an IVP:
n 1
n 1
o y  t0   y0 , y  t0   y0 , y    t0   y0 
Theorem 4.1.1: Existence and Uniqueness Theorem
(for nth-Order Linear Differential Equations)
If the functions where p1 , , pn , and g are continuous on the open interval I, then exists exactly
one solution y    t  of the differential equation
dny
d n1 y
dy
 p1  t  n 1   pn 1  t   pn  t  y  g  t  that also satisfied the initial conditions
n
dt
dt
dt
 n 1
y  t0   y0 , y  t0   y0 , y
t0   y0 n1 . This solution exists throughout the interval I.
Practice:
4
1. Determine an interval in which the solution of t  t  1 y    et y  4t 2 y  t is sure to
exist.
ODE Lecture Notes
Section 4.1
Page 2 of 6
The Homogeneous Equation:
L  y   y  n  p1  t  y  n1   pn1  t  y  pn  t  y  0
Notes:
 If the functions y1 , y2 ,
, yn are solutions, then so is a linear combination of them,
y  t   c1 y1  t   c2 y2 t  

To satisfy the initial conditions, we get n equations in n unknowns:
c1 y1  t0   c2 y2  t0    cn yn  t0   y0
c1 y1  t0   c2 y2  t0  
c1 y1 n 1  t0   c2 y2 n 1  t0  

yn  
 cn yn n 1  t0   y0 n 1
y1
y2
yn
y1 n 1
y2 n 1
yn n 1
, pn are continuous on the open interval I, if the functions
n
n 1
, yn are solutions of y   p1  t  y   
W  y1 , y2 ,
Notes:
 y  t   c1 y1  t   c2 y2 t  
y
 n
 pn1  t  y  pn  t  y  0 , and if
yn   0 for at least one point in I, then every solution of the differential equation can
be written as a linear combination of y1 , y2 ,

0
Note that a slightly modified form of Abel’s Theorem still applies:
W  y1 , y2 , yn  t   c exp    p1  t  dt 
Theorem 4.1.2:
If the functions where p1 ,
y1 , y2 ,
 cn yn  t0   y0
This system will have a solution for c1 , c2 , , cn provided the determinant of the matrix of
coefficients is not zero (i.e., Cramer’s Rule again). In other words, the Wronskian is
nonzero, just like for second-order equations.
y1
y2
yn
W  y1 , y2 ,

 cn yn t 
 p1  t  y
y1 , y2 ,
 n 1

, yn .
 cn yn t  is called the general solution of
 pn1  t  y  pn  t  y  0 .
, yn are said to form a fundamental set of solutions.
ODE Lecture Notes
Section 4.1
Page 3 of 6
Practice:
2. Verify that y1  t   1, y2  t   t , y3  t   cos  t  , y4 t   sin t  are solutions of
y    y  0 , and compute their Wronskian.
4
ODE Lecture Notes
Section 4.1
Page 4 of 6
Linear Dependence and Independence:
 The functions f1 , f 2 , , f n are said to be linearly dependent on an interval I if there exist
constants k1 , k2 ,

, kn , NOT ALL ZERO, such that k1 f1  t   k2 f 2 t  
FOR ALL t  I .
The functions f1 , f 2 ,
dependent there.
kn f n  t   0
, f n are said to be linearly independent on I if they are not linearly
Practice:
3. Determine if f1  t   1, f 2  t   t , f3  t   t 2 are linearly dependent or independent on
  t   . If they are dependent, write a linear relationship between them.
ODE Lecture Notes
Section 4.1
Page 5 of 6
4. Determine if f1  t   1, f 2 t   t , f3 t   2  t are linearly dependent or independent on
  t   . If they are dependent, write a linear relationship between them.
ODE Lecture Notes
Section 4.1
Theorem 4.1.3:
If y1 , y2 , , yn is a fundamental set of solutions to y n  p1  t  y  n1 
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 pn1  t  y  pn  t  y  0
on an interval I, then y1 , y2 , , yn are linearly independent on I. Conversely, if y1 , y2 , , yn are
linearly independent solutions of the equation, then they form a fundamental set of solutions on
I.
The Nonhomogeneous Equation:
L  y   y n  p1  t  y  n1   pn1  t  y  pn  t  y  g  t 
Notes:
 If Y1(t) and Y2(t) are solutions of the nonhomogeneous equation, then
L Y1  Y2   t   L Y1  t   L Y2  t   g t   g t   0



I.e., the difference of any two solutions of the nonhomogeneous equation is a solution of
the homogeneous equation.
So, the general solution of the nonhomogeneous equation is:
y  t   c1 y1  t   c2 y2  t    cn yn t   Y t  , where Y(t) is a particular solution of the
nonhomogeneous equation.
We will see that the methods of undetermined coefficients and reduction of order can be
extended from second-order equations to nth-order equations.