General Theory of Homogeneous and Non-Homogeneous linear ODEs - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


 Basic Theory of Linear O.D.E.’s  
 
Definition: A linear O.D.E. of order n, in the independent variable x and the dependent 
variable y is an equation that can be expressed in the form: 
                    a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
We will assume that a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions 
on the interval [a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
 
Remark: A linear ODE satisfies the following conditions: 
1- The dependent variable y and its derivatives occur to the first degree only. 
2- No products of y and/or any of its derivatives appear in the equation. 
3- No transcendental functions of y and/or its derivatives occur. 
 
Definition: Equations that not satisfy the above conditions are called nonlinear. 
 
Remark:  
1) The functions a
0
(x), a
1
(x), …. , a
n
(x) are called the coefficients of the equation.  
2) The right-hand member b(x) is called the non-homogeneous term. 
3) If b(x) = 0, then the equation reduces to  
                           a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
and it is called a homogeneous linear differential equation (H.L.D.E.). 
 
Example: 
1) 
d
3
y
dx
3
!cos(x)
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
 is a linear differential equation of order 3, 
where a
0
(x) = 1, a
2
(x) = - cos(x), a
3
(x) = 3x, a
4
(x) = x
3
 and b(x) = e
x
, they are all 
continuous function on 
 
! . 
 
Definition: If a
0
, a
1
, …. , a
n
 are all constants, then we have a linear differential equation 
with constant coefficients, otherwise we have an equation with variable coefficients. 
 
Example: 
1) 3y’’’ + 2y’’ – 4y’ – 7y = sin x is an equation with constant coefficients. 
2) 2y’’ + xy’ – e
x
 y = 0 is an equation with variable coefficients. 
 
Theorem: Let 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
with a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions on the interval 
[a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
Let x
0
 be a point on the interval [a, b] and let c
0
, c
1
, c
n-1
 be n arbitrary constants. 
Conclusion: 
There exists a unique function f(x) defined on [a, b] such that f(x) is a solution of the 
linear differential equation and satisfies 
                            f(x
0
) = c
0
, f’(x
0
) = c
1
, f’’(x
0
) = c
2
, … , f
(n-1)
(x
0
) = c
n-1
. 
Page 2


 Basic Theory of Linear O.D.E.’s  
 
Definition: A linear O.D.E. of order n, in the independent variable x and the dependent 
variable y is an equation that can be expressed in the form: 
                    a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
We will assume that a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions 
on the interval [a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
 
Remark: A linear ODE satisfies the following conditions: 
1- The dependent variable y and its derivatives occur to the first degree only. 
2- No products of y and/or any of its derivatives appear in the equation. 
3- No transcendental functions of y and/or its derivatives occur. 
 
Definition: Equations that not satisfy the above conditions are called nonlinear. 
 
Remark:  
1) The functions a
0
(x), a
1
(x), …. , a
n
(x) are called the coefficients of the equation.  
2) The right-hand member b(x) is called the non-homogeneous term. 
3) If b(x) = 0, then the equation reduces to  
                           a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
and it is called a homogeneous linear differential equation (H.L.D.E.). 
 
Example: 
1) 
d
3
y
dx
3
!cos(x)
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
 is a linear differential equation of order 3, 
where a
0
(x) = 1, a
2
(x) = - cos(x), a
3
(x) = 3x, a
4
(x) = x
3
 and b(x) = e
x
, they are all 
continuous function on 
 
! . 
 
Definition: If a
0
, a
1
, …. , a
n
 are all constants, then we have a linear differential equation 
with constant coefficients, otherwise we have an equation with variable coefficients. 
 
Example: 
1) 3y’’’ + 2y’’ – 4y’ – 7y = sin x is an equation with constant coefficients. 
2) 2y’’ + xy’ – e
x
 y = 0 is an equation with variable coefficients. 
 
Theorem: Let 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
with a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions on the interval 
[a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
Let x
0
 be a point on the interval [a, b] and let c
0
, c
1
, c
n-1
 be n arbitrary constants. 
Conclusion: 
There exists a unique function f(x) defined on [a, b] such that f(x) is a solution of the 
linear differential equation and satisfies 
                            f(x
0
) = c
0
, f’(x
0
) = c
1
, f’’(x
0
) = c
2
, … , f
(n-1)
(x
0
) = c
n-1
. 
Remark: The theorem extends the I.V.P. discussed for first order differential equations 
to linear differential equations of order n. 
 
Example: 
Consider the I.V.P. 
                             
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
y(1)=2
y'(1)= !5
"
#
$
$
%
$
$
 
a
0
(x) = 1, a
1
(x) = 3x, a
3
(x) = x
3
, and b(x) = e
x
, they are continuous functions everywhere.  
We have a linear differential equation order 2.  The real numbers c
0
 = 2 and c
1
 = -5. 
Then, according to the theorem there exists an unique function f(x) continuous 
everywhere that is the solution of the equation and f(1) = 2 and f(x) = -5. 
 
Corollary: Given the H.L.D.E. 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
with a
0
(x), a
1
(x), …. , a
n
(x) are continuous real-valued functions on the interval [a, b] and 
a
0
(x) ? 0 for at least one x in [a, b]. 
Let f(x) be a solution of the H.L.D.E. such that f(x
0
) = 0, f’(x
0
) = 0, f’’(x
0
) = 0, … ,  
f
(n-1)
(x
0
) = 0  where x
0
 is in [a, b],  
Conclusion: 
                                           f(x) = 0 for all x in [a, b]. 
 
 
Example: 
                                       y’’’ + 2y’’ + 4xy’ + x
2
y = 0 
                                           y(2) = y’(2) = y’’(2) = 0 
The unique solution that satisfies the I.V.P. is f(x) = 0 
 
Definition: Given k functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
k
(x), and k 
constants, c
1
, c
2
, … , c
k
.  The expression  
                                                c
1
f
1
(x) + c
2
f
2
(x) + … + c
k
f
k
(x)  
is called a linear combination of f
1
(x), f
2
(x), … , f
k
(x). 
 
Basic Theorem on Homogeneous Linear Differential Equations 
Given the n-th order H.L.D.E. 
                     a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
let f
1
(x), f
2
(x), … , f
k
(x) be k solutions of the H.L.D.E. on the interval [a, b]. 
Conclusion: 
The linear combination    c
1
f
1
(x) + c
2
f
2
(x) + … + c
k
f
k
(x)  is a solution of H.L.D.E. on          
[a, b] where c
1
, c
2
, … , c
k
 are k arbitrary constants. 
 
Example:  
Page 3


 Basic Theory of Linear O.D.E.’s  
 
Definition: A linear O.D.E. of order n, in the independent variable x and the dependent 
variable y is an equation that can be expressed in the form: 
                    a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
We will assume that a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions 
on the interval [a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
 
Remark: A linear ODE satisfies the following conditions: 
1- The dependent variable y and its derivatives occur to the first degree only. 
2- No products of y and/or any of its derivatives appear in the equation. 
3- No transcendental functions of y and/or its derivatives occur. 
 
Definition: Equations that not satisfy the above conditions are called nonlinear. 
 
Remark:  
1) The functions a
0
(x), a
1
(x), …. , a
n
(x) are called the coefficients of the equation.  
2) The right-hand member b(x) is called the non-homogeneous term. 
3) If b(x) = 0, then the equation reduces to  
                           a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
and it is called a homogeneous linear differential equation (H.L.D.E.). 
 
Example: 
1) 
d
3
y
dx
3
!cos(x)
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
 is a linear differential equation of order 3, 
where a
0
(x) = 1, a
2
(x) = - cos(x), a
3
(x) = 3x, a
4
(x) = x
3
 and b(x) = e
x
, they are all 
continuous function on 
 
! . 
 
Definition: If a
0
, a
1
, …. , a
n
 are all constants, then we have a linear differential equation 
with constant coefficients, otherwise we have an equation with variable coefficients. 
 
Example: 
1) 3y’’’ + 2y’’ – 4y’ – 7y = sin x is an equation with constant coefficients. 
2) 2y’’ + xy’ – e
x
 y = 0 is an equation with variable coefficients. 
 
Theorem: Let 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
with a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions on the interval 
[a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
Let x
0
 be a point on the interval [a, b] and let c
0
, c
1
, c
n-1
 be n arbitrary constants. 
Conclusion: 
There exists a unique function f(x) defined on [a, b] such that f(x) is a solution of the 
linear differential equation and satisfies 
                            f(x
0
) = c
0
, f’(x
0
) = c
1
, f’’(x
0
) = c
2
, … , f
(n-1)
(x
0
) = c
n-1
. 
Remark: The theorem extends the I.V.P. discussed for first order differential equations 
to linear differential equations of order n. 
 
Example: 
Consider the I.V.P. 
                             
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
y(1)=2
y'(1)= !5
"
#
$
$
%
$
$
 
a
0
(x) = 1, a
1
(x) = 3x, a
3
(x) = x
3
, and b(x) = e
x
, they are continuous functions everywhere.  
We have a linear differential equation order 2.  The real numbers c
0
 = 2 and c
1
 = -5. 
Then, according to the theorem there exists an unique function f(x) continuous 
everywhere that is the solution of the equation and f(1) = 2 and f(x) = -5. 
 
Corollary: Given the H.L.D.E. 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
with a
0
(x), a
1
(x), …. , a
n
(x) are continuous real-valued functions on the interval [a, b] and 
a
0
(x) ? 0 for at least one x in [a, b]. 
Let f(x) be a solution of the H.L.D.E. such that f(x
0
) = 0, f’(x
0
) = 0, f’’(x
0
) = 0, … ,  
f
(n-1)
(x
0
) = 0  where x
0
 is in [a, b],  
Conclusion: 
                                           f(x) = 0 for all x in [a, b]. 
 
 
Example: 
                                       y’’’ + 2y’’ + 4xy’ + x
2
y = 0 
                                           y(2) = y’(2) = y’’(2) = 0 
The unique solution that satisfies the I.V.P. is f(x) = 0 
 
Definition: Given k functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
k
(x), and k 
constants, c
1
, c
2
, … , c
k
.  The expression  
                                                c
1
f
1
(x) + c
2
f
2
(x) + … + c
k
f
k
(x)  
is called a linear combination of f
1
(x), f
2
(x), … , f
k
(x). 
 
Basic Theorem on Homogeneous Linear Differential Equations 
Given the n-th order H.L.D.E. 
                     a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
let f
1
(x), f
2
(x), … , f
k
(x) be k solutions of the H.L.D.E. on the interval [a, b]. 
Conclusion: 
The linear combination    c
1
f
1
(x) + c
2
f
2
(x) + … + c
k
f
k
(x)  is a solution of H.L.D.E. on          
[a, b] where c
1
, c
2
, … , c
k
 are k arbitrary constants. 
 
Example:  
Given the H.L.D.E.  
d
2
y
dx
2
+y=0 ,  
The functions f
1
(x) = sin x and f
2
(x) = cos x are solutions of the equation,  since taking 
the second derivative of the function f
1
''
(x)=!sin!x and substituting into the equation, 
we get:  –sin x + sin x = 0.  Similarly for the other function. 
Then g(x) = c
1
 sin x + c
2
 cos x is a solution of the equation where c
1
 and c
2
 are any 
arbitrary constant: g’’(x) = - c
1
 sinx – c
2
 cos x, substituting into the equation we get 
- c
1
 sinx – c
2
 cos x + c
1
 sinx + c
2
 cos x = 0. 
 
Definition: n functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
n
(x) are called 
linearly dependent on [a, b] if there exist constants c
1
, c
2
, … , c
n
, not all of them zero, 
such that the linear combination  
                                                c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) = 0 
for all x in [a, b]. 
 
Example: 
1) The functions f
1
(x) = x
2
 and f
2
(x) = 4x
2
 are linearly dependent on [-1, 1] since  
there exist constants c
1
 = - 4 and c
2
 = 1 such that c
1
f
1
(x) + c
2
f
2
(x) = -4x
2
 + 4x
2
 = 0 for all 
x in [-1, 1]. 
 
2) The functions f
1
(x) = e
x
 , f
2
(x) = - e
x
 and f
3
(x) = 8e
x
 are linearly dependent on [-10, 10] 
since there exist constants c
1
 = 5, c
2
 = 1 and c
3
 = 1 such that  
          c
1
f
1
(x) + c
2
f
2
(x) + c
3
f
3
(x) = 5e
x
 – e
x
 
+ 4e
x
 = 0 for all x in [-10, 10]. 
 
Definition: n functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
n
(x) are called 
linearly independent on [a, b] if they are NOT linear dependent on [a, b]. 
 
This means that f
1
(x), f
2
(x), … , f
n
(x) are linearly independent on [a, b] if the relation 
                        c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) = 0  for all x in [a, b]  
implies that c
1
 = c
2
 = … = c
n
 = 0. 
 
Example: 
The functions f
1
(x) = x
2
 , f
2
(x) = x
3
 and f
3
(x) = x
5
 are linearly independent on [-5, 5] since 
the relation  
                 c
1
f
1
(x) + c
2
f
2
(x) + c
3
f
3
(x) = c
1
x
2
 + c
2
x
3
 + c
3
x
5
 = 0 for all x in [-5, 5] 
if and only if  c
1
 = c
2
 = c
3
 = 0. 
 
Theorem: The n-th order H.L.D.E. 
                     a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
always possesses n solutions f
1
(x), f
2
(x), … , f
n
(x) that are linearly independent. 
Furthermore, if f
1
(x), f
2
(x), … , f
n
(x) are n linearly independent solutions of the equation, 
then every solution f(x) of the equation can be expressed as a linear combination 
                                   f(x) = c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) 
of those n solutions by a proper choice of constants c
1
, c
2
, … , c
n
. 
 
Page 4


 Basic Theory of Linear O.D.E.’s  
 
Definition: A linear O.D.E. of order n, in the independent variable x and the dependent 
variable y is an equation that can be expressed in the form: 
                    a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
We will assume that a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions 
on the interval [a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
 
Remark: A linear ODE satisfies the following conditions: 
1- The dependent variable y and its derivatives occur to the first degree only. 
2- No products of y and/or any of its derivatives appear in the equation. 
3- No transcendental functions of y and/or its derivatives occur. 
 
Definition: Equations that not satisfy the above conditions are called nonlinear. 
 
Remark:  
1) The functions a
0
(x), a
1
(x), …. , a
n
(x) are called the coefficients of the equation.  
2) The right-hand member b(x) is called the non-homogeneous term. 
3) If b(x) = 0, then the equation reduces to  
                           a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
and it is called a homogeneous linear differential equation (H.L.D.E.). 
 
Example: 
1) 
d
3
y
dx
3
!cos(x)
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
 is a linear differential equation of order 3, 
where a
0
(x) = 1, a
2
(x) = - cos(x), a
3
(x) = 3x, a
4
(x) = x
3
 and b(x) = e
x
, they are all 
continuous function on 
 
! . 
 
Definition: If a
0
, a
1
, …. , a
n
 are all constants, then we have a linear differential equation 
with constant coefficients, otherwise we have an equation with variable coefficients. 
 
Example: 
1) 3y’’’ + 2y’’ – 4y’ – 7y = sin x is an equation with constant coefficients. 
2) 2y’’ + xy’ – e
x
 y = 0 is an equation with variable coefficients. 
 
Theorem: Let 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
with a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions on the interval 
[a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
Let x
0
 be a point on the interval [a, b] and let c
0
, c
1
, c
n-1
 be n arbitrary constants. 
Conclusion: 
There exists a unique function f(x) defined on [a, b] such that f(x) is a solution of the 
linear differential equation and satisfies 
                            f(x
0
) = c
0
, f’(x
0
) = c
1
, f’’(x
0
) = c
2
, … , f
(n-1)
(x
0
) = c
n-1
. 
Remark: The theorem extends the I.V.P. discussed for first order differential equations 
to linear differential equations of order n. 
 
Example: 
Consider the I.V.P. 
                             
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
y(1)=2
y'(1)= !5
"
#
$
$
%
$
$
 
a
0
(x) = 1, a
1
(x) = 3x, a
3
(x) = x
3
, and b(x) = e
x
, they are continuous functions everywhere.  
We have a linear differential equation order 2.  The real numbers c
0
 = 2 and c
1
 = -5. 
Then, according to the theorem there exists an unique function f(x) continuous 
everywhere that is the solution of the equation and f(1) = 2 and f(x) = -5. 
 
Corollary: Given the H.L.D.E. 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
with a
0
(x), a
1
(x), …. , a
n
(x) are continuous real-valued functions on the interval [a, b] and 
a
0
(x) ? 0 for at least one x in [a, b]. 
Let f(x) be a solution of the H.L.D.E. such that f(x
0
) = 0, f’(x
0
) = 0, f’’(x
0
) = 0, … ,  
f
(n-1)
(x
0
) = 0  where x
0
 is in [a, b],  
Conclusion: 
                                           f(x) = 0 for all x in [a, b]. 
 
 
Example: 
                                       y’’’ + 2y’’ + 4xy’ + x
2
y = 0 
                                           y(2) = y’(2) = y’’(2) = 0 
The unique solution that satisfies the I.V.P. is f(x) = 0 
 
Definition: Given k functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
k
(x), and k 
constants, c
1
, c
2
, … , c
k
.  The expression  
                                                c
1
f
1
(x) + c
2
f
2
(x) + … + c
k
f
k
(x)  
is called a linear combination of f
1
(x), f
2
(x), … , f
k
(x). 
 
Basic Theorem on Homogeneous Linear Differential Equations 
Given the n-th order H.L.D.E. 
                     a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
let f
1
(x), f
2
(x), … , f
k
(x) be k solutions of the H.L.D.E. on the interval [a, b]. 
Conclusion: 
The linear combination    c
1
f
1
(x) + c
2
f
2
(x) + … + c
k
f
k
(x)  is a solution of H.L.D.E. on          
[a, b] where c
1
, c
2
, … , c
k
 are k arbitrary constants. 
 
Example:  
Given the H.L.D.E.  
d
2
y
dx
2
+y=0 ,  
The functions f
1
(x) = sin x and f
2
(x) = cos x are solutions of the equation,  since taking 
the second derivative of the function f
1
''
(x)=!sin!x and substituting into the equation, 
we get:  –sin x + sin x = 0.  Similarly for the other function. 
Then g(x) = c
1
 sin x + c
2
 cos x is a solution of the equation where c
1
 and c
2
 are any 
arbitrary constant: g’’(x) = - c
1
 sinx – c
2
 cos x, substituting into the equation we get 
- c
1
 sinx – c
2
 cos x + c
1
 sinx + c
2
 cos x = 0. 
 
Definition: n functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
n
(x) are called 
linearly dependent on [a, b] if there exist constants c
1
, c
2
, … , c
n
, not all of them zero, 
such that the linear combination  
                                                c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) = 0 
for all x in [a, b]. 
 
Example: 
1) The functions f
1
(x) = x
2
 and f
2
(x) = 4x
2
 are linearly dependent on [-1, 1] since  
there exist constants c
1
 = - 4 and c
2
 = 1 such that c
1
f
1
(x) + c
2
f
2
(x) = -4x
2
 + 4x
2
 = 0 for all 
x in [-1, 1]. 
 
2) The functions f
1
(x) = e
x
 , f
2
(x) = - e
x
 and f
3
(x) = 8e
x
 are linearly dependent on [-10, 10] 
since there exist constants c
1
 = 5, c
2
 = 1 and c
3
 = 1 such that  
          c
1
f
1
(x) + c
2
f
2
(x) + c
3
f
3
(x) = 5e
x
 – e
x
 
+ 4e
x
 = 0 for all x in [-10, 10]. 
 
Definition: n functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
n
(x) are called 
linearly independent on [a, b] if they are NOT linear dependent on [a, b]. 
 
This means that f
1
(x), f
2
(x), … , f
n
(x) are linearly independent on [a, b] if the relation 
                        c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) = 0  for all x in [a, b]  
implies that c
1
 = c
2
 = … = c
n
 = 0. 
 
Example: 
The functions f
1
(x) = x
2
 , f
2
(x) = x
3
 and f
3
(x) = x
5
 are linearly independent on [-5, 5] since 
the relation  
                 c
1
f
1
(x) + c
2
f
2
(x) + c
3
f
3
(x) = c
1
x
2
 + c
2
x
3
 + c
3
x
5
 = 0 for all x in [-5, 5] 
if and only if  c
1
 = c
2
 = c
3
 = 0. 
 
Theorem: The n-th order H.L.D.E. 
                     a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
always possesses n solutions f
1
(x), f
2
(x), … , f
n
(x) that are linearly independent. 
Furthermore, if f
1
(x), f
2
(x), … , f
n
(x) are n linearly independent solutions of the equation, 
then every solution f(x) of the equation can be expressed as a linear combination 
                                   f(x) = c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) 
of those n solutions by a proper choice of constants c
1
, c
2
, … , c
n
. 
 
Example: 
Given the 2
nd
 order H.L.D.E.   y’’ + y = 0,  
the functions f
1
(x) = sin x and f
2
(x) = cos x are two linearly independent solutions of the 
equation on (-8, 8). 
If f(x) is any other solution of the equation then f(x) = c
1
 sin x + c
2
 cos x, for particular 
values of the constants c
1
 and c
2
. 
For example f(x) = cos(x – p/3) is a solution of the equation, 
Since cos(x – p/3) = cos x cos(p/3) + sin x sin(p/3) = ½ cos x + 
3
2
 sin x, then  
                                f(x) = 
3
2
 sin x + ½ cos x  = 
3
2
 f
1
(x) + ½ f
2
(x) 
 
Definition: If f
1
(x), f
2
(x), … , f
n
(x) are n linearly independent solutions of the H.L.D.E. 
                          a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
on [a, b],  then the set { f
1
(x), f
2
(x), … , f
n
(x)} is called the fundamental set of solutions 
of the equation. 
The function f(x) = c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) where c
1
, c
2
, … , c
n
 are arbitrary 
constants is called a general solution of the equation. 
 
Example: 
Given the 3
rd
 order H.L.D.E.   y’’’ - 2y’’ – y’ + 2y = 0,  
the functions f
1
(x) = e
x
, f
2
(x) = e
-x
, and f
3
(x) = e
2x
  are three linearly independent solutions 
of the equation on (-8, 8). 
The set {e
x
, e
-x
, e
2x
} is the fundamental set of solutions. 
Then f(x) = c
1
 ex+ c
2
 e-x + c
3
 e
2x
, where c
1
, c
2
 and c
3
 are arbitrary constants is a general 
solution of the equation. 
 
Definition: Let f
1
(x), f
2
(x), … , f
n
(x) be n real-valued functions defined on [a, b] such that 
the (n -1)-st derivative of each function exists in [a, b]. 
The determinant 
                  
 
W f
1
,f
2
,...,f
n
( )(x)=
f
1
(x) f
2
(x) ! f
n
(x)
f
1
'
(x) f
2
'
(x) ! f
n
'
(x)
! ! ! !
f
1
(n!1)
(x) f
2
(n!1)
(x) ! f
n
(n!1)
(x)
 
is called the Wronskian of f
1
(x), f
2
(x), … , f
n
(x). 
 
Remark: The Wronskian is also a function of x. 
 
Theorem: The n solutions f
1
(x), f
2
(x), … , f
n
(x)  of the n-th H.L.D.E. 
                          a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
are linearly independent on [a, b] if and only if  W(f
1
, f
2
, … , f
n
) (x) ? 0 for at least one x 
on [a, b]. 
 
Example:  
Page 5


 Basic Theory of Linear O.D.E.’s  
 
Definition: A linear O.D.E. of order n, in the independent variable x and the dependent 
variable y is an equation that can be expressed in the form: 
                    a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
We will assume that a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions 
on the interval [a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
 
Remark: A linear ODE satisfies the following conditions: 
1- The dependent variable y and its derivatives occur to the first degree only. 
2- No products of y and/or any of its derivatives appear in the equation. 
3- No transcendental functions of y and/or its derivatives occur. 
 
Definition: Equations that not satisfy the above conditions are called nonlinear. 
 
Remark:  
1) The functions a
0
(x), a
1
(x), …. , a
n
(x) are called the coefficients of the equation.  
2) The right-hand member b(x) is called the non-homogeneous term. 
3) If b(x) = 0, then the equation reduces to  
                           a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
and it is called a homogeneous linear differential equation (H.L.D.E.). 
 
Example: 
1) 
d
3
y
dx
3
!cos(x)
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
 is a linear differential equation of order 3, 
where a
0
(x) = 1, a
2
(x) = - cos(x), a
3
(x) = 3x, a
4
(x) = x
3
 and b(x) = e
x
, they are all 
continuous function on 
 
! . 
 
Definition: If a
0
, a
1
, …. , a
n
 are all constants, then we have a linear differential equation 
with constant coefficients, otherwise we have an equation with variable coefficients. 
 
Example: 
1) 3y’’’ + 2y’’ – 4y’ – 7y = sin x is an equation with constant coefficients. 
2) 2y’’ + xy’ – e
x
 y = 0 is an equation with variable coefficients. 
 
Theorem: Let 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
with a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions on the interval 
[a, b] and a
0
(x) ? 0 for at least one x in [a, b]. 
Let x
0
 be a point on the interval [a, b] and let c
0
, c
1
, c
n-1
 be n arbitrary constants. 
Conclusion: 
There exists a unique function f(x) defined on [a, b] such that f(x) is a solution of the 
linear differential equation and satisfies 
                            f(x
0
) = c
0
, f’(x
0
) = c
1
, f’’(x
0
) = c
2
, … , f
(n-1)
(x
0
) = c
n-1
. 
Remark: The theorem extends the I.V.P. discussed for first order differential equations 
to linear differential equations of order n. 
 
Example: 
Consider the I.V.P. 
                             
d
2
y
dx
2
+3x
dy
dx
+x
3
y=e
x
y(1)=2
y'(1)= !5
"
#
$
$
%
$
$
 
a
0
(x) = 1, a
1
(x) = 3x, a
3
(x) = x
3
, and b(x) = e
x
, they are continuous functions everywhere.  
We have a linear differential equation order 2.  The real numbers c
0
 = 2 and c
1
 = -5. 
Then, according to the theorem there exists an unique function f(x) continuous 
everywhere that is the solution of the equation and f(1) = 2 and f(x) = -5. 
 
Corollary: Given the H.L.D.E. 
                       a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
with a
0
(x), a
1
(x), …. , a
n
(x) are continuous real-valued functions on the interval [a, b] and 
a
0
(x) ? 0 for at least one x in [a, b]. 
Let f(x) be a solution of the H.L.D.E. such that f(x
0
) = 0, f’(x
0
) = 0, f’’(x
0
) = 0, … ,  
f
(n-1)
(x
0
) = 0  where x
0
 is in [a, b],  
Conclusion: 
                                           f(x) = 0 for all x in [a, b]. 
 
 
Example: 
                                       y’’’ + 2y’’ + 4xy’ + x
2
y = 0 
                                           y(2) = y’(2) = y’’(2) = 0 
The unique solution that satisfies the I.V.P. is f(x) = 0 
 
Definition: Given k functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
k
(x), and k 
constants, c
1
, c
2
, … , c
k
.  The expression  
                                                c
1
f
1
(x) + c
2
f
2
(x) + … + c
k
f
k
(x)  
is called a linear combination of f
1
(x), f
2
(x), … , f
k
(x). 
 
Basic Theorem on Homogeneous Linear Differential Equations 
Given the n-th order H.L.D.E. 
                     a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
let f
1
(x), f
2
(x), … , f
k
(x) be k solutions of the H.L.D.E. on the interval [a, b]. 
Conclusion: 
The linear combination    c
1
f
1
(x) + c
2
f
2
(x) + … + c
k
f
k
(x)  is a solution of H.L.D.E. on          
[a, b] where c
1
, c
2
, … , c
k
 are k arbitrary constants. 
 
Example:  
Given the H.L.D.E.  
d
2
y
dx
2
+y=0 ,  
The functions f
1
(x) = sin x and f
2
(x) = cos x are solutions of the equation,  since taking 
the second derivative of the function f
1
''
(x)=!sin!x and substituting into the equation, 
we get:  –sin x + sin x = 0.  Similarly for the other function. 
Then g(x) = c
1
 sin x + c
2
 cos x is a solution of the equation where c
1
 and c
2
 are any 
arbitrary constant: g’’(x) = - c
1
 sinx – c
2
 cos x, substituting into the equation we get 
- c
1
 sinx – c
2
 cos x + c
1
 sinx + c
2
 cos x = 0. 
 
Definition: n functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
n
(x) are called 
linearly dependent on [a, b] if there exist constants c
1
, c
2
, … , c
n
, not all of them zero, 
such that the linear combination  
                                                c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) = 0 
for all x in [a, b]. 
 
Example: 
1) The functions f
1
(x) = x
2
 and f
2
(x) = 4x
2
 are linearly dependent on [-1, 1] since  
there exist constants c
1
 = - 4 and c
2
 = 1 such that c
1
f
1
(x) + c
2
f
2
(x) = -4x
2
 + 4x
2
 = 0 for all 
x in [-1, 1]. 
 
2) The functions f
1
(x) = e
x
 , f
2
(x) = - e
x
 and f
3
(x) = 8e
x
 are linearly dependent on [-10, 10] 
since there exist constants c
1
 = 5, c
2
 = 1 and c
3
 = 1 such that  
          c
1
f
1
(x) + c
2
f
2
(x) + c
3
f
3
(x) = 5e
x
 – e
x
 
+ 4e
x
 = 0 for all x in [-10, 10]. 
 
Definition: n functions defined on an interval [a, b], f
1
(x), f
2
(x), … , f
n
(x) are called 
linearly independent on [a, b] if they are NOT linear dependent on [a, b]. 
 
This means that f
1
(x), f
2
(x), … , f
n
(x) are linearly independent on [a, b] if the relation 
                        c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) = 0  for all x in [a, b]  
implies that c
1
 = c
2
 = … = c
n
 = 0. 
 
Example: 
The functions f
1
(x) = x
2
 , f
2
(x) = x
3
 and f
3
(x) = x
5
 are linearly independent on [-5, 5] since 
the relation  
                 c
1
f
1
(x) + c
2
f
2
(x) + c
3
f
3
(x) = c
1
x
2
 + c
2
x
3
 + c
3
x
5
 = 0 for all x in [-5, 5] 
if and only if  c
1
 = c
2
 = c
3
 = 0. 
 
Theorem: The n-th order H.L.D.E. 
                     a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
always possesses n solutions f
1
(x), f
2
(x), … , f
n
(x) that are linearly independent. 
Furthermore, if f
1
(x), f
2
(x), … , f
n
(x) are n linearly independent solutions of the equation, 
then every solution f(x) of the equation can be expressed as a linear combination 
                                   f(x) = c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) 
of those n solutions by a proper choice of constants c
1
, c
2
, … , c
n
. 
 
Example: 
Given the 2
nd
 order H.L.D.E.   y’’ + y = 0,  
the functions f
1
(x) = sin x and f
2
(x) = cos x are two linearly independent solutions of the 
equation on (-8, 8). 
If f(x) is any other solution of the equation then f(x) = c
1
 sin x + c
2
 cos x, for particular 
values of the constants c
1
 and c
2
. 
For example f(x) = cos(x – p/3) is a solution of the equation, 
Since cos(x – p/3) = cos x cos(p/3) + sin x sin(p/3) = ½ cos x + 
3
2
 sin x, then  
                                f(x) = 
3
2
 sin x + ½ cos x  = 
3
2
 f
1
(x) + ½ f
2
(x) 
 
Definition: If f
1
(x), f
2
(x), … , f
n
(x) are n linearly independent solutions of the H.L.D.E. 
                          a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
on [a, b],  then the set { f
1
(x), f
2
(x), … , f
n
(x)} is called the fundamental set of solutions 
of the equation. 
The function f(x) = c
1
f
1
(x) + c
2
f
2
(x) + … + c
n
f
n
(x) where c
1
, c
2
, … , c
n
 are arbitrary 
constants is called a general solution of the equation. 
 
Example: 
Given the 3
rd
 order H.L.D.E.   y’’’ - 2y’’ – y’ + 2y = 0,  
the functions f
1
(x) = e
x
, f
2
(x) = e
-x
, and f
3
(x) = e
2x
  are three linearly independent solutions 
of the equation on (-8, 8). 
The set {e
x
, e
-x
, e
2x
} is the fundamental set of solutions. 
Then f(x) = c
1
 ex+ c
2
 e-x + c
3
 e
2x
, where c
1
, c
2
 and c
3
 are arbitrary constants is a general 
solution of the equation. 
 
Definition: Let f
1
(x), f
2
(x), … , f
n
(x) be n real-valued functions defined on [a, b] such that 
the (n -1)-st derivative of each function exists in [a, b]. 
The determinant 
                  
 
W f
1
,f
2
,...,f
n
( )(x)=
f
1
(x) f
2
(x) ! f
n
(x)
f
1
'
(x) f
2
'
(x) ! f
n
'
(x)
! ! ! !
f
1
(n!1)
(x) f
2
(n!1)
(x) ! f
n
(n!1)
(x)
 
is called the Wronskian of f
1
(x), f
2
(x), … , f
n
(x). 
 
Remark: The Wronskian is also a function of x. 
 
Theorem: The n solutions f
1
(x), f
2
(x), … , f
n
(x)  of the n-th H.L.D.E. 
                          a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
are linearly independent on [a, b] if and only if  W(f
1
, f
2
, … , f
n
) (x) ? 0 for at least one x 
on [a, b]. 
 
Example:  
1) Show that the functions f
1
(x) = e
x
, f
2
(x) = e
-x
, and f
3
(x) = e
2x
 are three linearly 
independent solutions of the 3
rd
 order H.L.D.E. y’’’ - 2y’’ – y’ + 2y = 0, on (-8, 8). 
Let’s compute the Wronskian 
W e
x
,e
!x
,e
2x
( )
=
e
x
e
!x
e
2x
e
x
!e
!x
2e
2x
e
x
e
!x
4e
2x
=e
x
e
!x
e
2x
1 1 1
1 !1 2
1 1 4
=e
2x
(!6)"0 for all x. 
 
2) Show that the functions f
1
(x) = x
2
, f
2
(x) = x
3
, and f
3
(x) = x 
-2
 are three linearly 
independent solutions of the 3
rd
 order H.L.D.E. x
3
y’’’ – 6xy’ + 12y = 0, on (-8, 8). 
Let’s compute the Wronskian 
W(x
2
,x
3
,x
!2
)=
x
2
x
3
x
!2
2x 3x
2
!2x
!3
2 6x 6x
!4
=x
2
3x
2
!2x
!3
6x 6x
!4
!x
3
2x !2x
!3
2 6x
!4
+x
!2
2x 3x
2
2 6x
=x
2
18x
!2
+12x
!2
( )
!x
3
12x
!3
+4x
!3
( )
+x
!2
12x
2
!6x
2
( )
=30!16+6 =20"0
 
 
The Non-Homogeneous Linear Differential Equation 
Given the n-th order non-homogeneous linear differential equation 
                   a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
then the n-th order H.L.D.E. 
                   a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 
is called the corresponding homogeneous equation. 
 
Remark: Every non-homogeneous differential equation has a corresponding 
homogeneous equation; we obtain it from the non-homogeneous replacing the non-
homogeneous term by zero. 
 
Theorem: Let v(x) be a solution of the non-homogeneous equation  
                  a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =b(x) 
Let u(x) be a solution of the corresponding homogeneous equation 
                    a
0
(x)y
n ( )
+a
1
(x)y
n!1 ( )
+.....+a
n!1
(x) " y +a
n
(x)y =0 . 
Conclusion: 
Then u(x) + v(x) is also a solution of the non-homogeneous equation. 
 
Example: 
Given the non-homogeneous equation  y’’ + y = x, 
The function v(x) = x is a solution of the equation. 
The corresponding homogeneous equation is y’’ + y = 0, and u(x) = sin x is a solution of 
this equation. 
Then v(x) + u(x) = x + sin x is also a solution of the non-homogeneous equation.