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4.1 Preliminary Theory – Linear Equations
4.1.1 Initial Value Problem (IVP)
An nth-order initial value problem (IVP) is given by:
Solve: an ( x) y ( n)  an 1 ( x) y ( n 1)  ...  a2 ( x) y   a1 ( x) y   a0 ( x) y  g ( x)
subject to the initial conditions:
y ( x0 )  y0 , y( x0 )  y1, ... , y ( n 1) ( x0 )  yn 1
Existence of a Unique Solution
Theorem 4.1.1
Let an ( x), an 1( x),..., a1( x), a0 ( x) and g ( x ) be continuous on an interval I, and let an ( x)  0
for every x in this interval. If x  x0 is any point in this interval, then a solution of the initialvalue problem exists on the interval and is unique.
Example:
1)
3 y ''' 5 y '' y ' 7 y  0 ,
2)
x2 y '' 2 xy ' 2 y  6 ,
y (1)  0, y '(1)  0, y ''(1)
y(0)  3, y '(0)  1
4.1.2 Homogeneous Equations
A homogeneous linear nth-order DE:
an ( x) y ( n)  an 1( x) y ( n 1)  ...  a2 ( x) y  a1( x) y   a0 ( x) y  0
A non-homogeneous linear nth-order DE:
an ( x) y ( n)  an 1 ( x) y ( n 1)  ...  a2 ( x) y   a1 ( x) y   a0 ( x) y  g ( x)
Examples:
y ''' 4 y '' 5 y '  0
(homogeneous)
y (4)  2 y ''' y ''  e x
(non-homogeneous)
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4.1 Preliminary Theory – Linear Equations
Superposition Principle – Homogeneous Equations
Theorem 4.1.2
Let y1, y2 ,..., yk be solutions of the homogeneous nth-order DE on any interval I. Then the linear
combination
y  c1 y1( x)  c2 y2 ( x)  ...  ck yk ( x),
where ci , i  1, 2,..., k are arbitrary constants, is also a solution on the interval.
Example: Let y1  e x and y2  e  x are the solutions of the homogeneous 2nd order DE:
y '' y  0 . By superposition principle, the linear combination
y  c1e x  c2e  x
is also a solution of the given DE.
Note:
 For a homogeneous linear DE, we can always obtain new solutions from known solutions
by multiplication by constants and by addition.
 The superposition principle does not hold for non-homogeneous linear DE as well as for
nonlinear DE.
Linear Dependence and Linear Independence
Definition 4.1.1
A set of functions f1( x), f 2 ( x),..., f n ( x) is said to be linearly dependent on an interval I if there
exist constants c1, c2 ,..., cn , not all zero, such that
c1 f1( x)  c2 f 2 ( x)  ...  cn f n ( x)  0
for every x in the interval.
In other words, the set of functions is said to be linearly independent if c1  c2  ...  cn  0
Example: Determine whether the functions are linear independent on the interval (, )
a)
f1 ( x)  x, f 2 ( x)  x 2 , f3 ( x)  x3
b)
f1 ( x)  5, f 2 ( x)  sin 2 x, f3 ( x)  cos 2 x
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4.1 Preliminary Theory – Linear Equations
Definition 4.1.2 (Wronskian)
Suppose each of the functions f1( x), f 2 ( x),..., f n ( x) possesses at least n – 1 derivatives. The
determinant
W ( f1, f 2 ,..., f n ) 
f1
f1
:
f1( n 1)
f2
f 2
:
...
...
:
f 2( n 1) ...
fn
f n
:
f n( n 1)
where the primes denote derivatives, is called the Wronskian of the functions.
Theorem 4.1.3
Let y1, y2 ,..., yn be n solutions of the homogeneous linear nth-order differential equation on an
interval I. Then the set of solutions is linearly independent on I if and only if
W ( y1, y2 ,..., yn )  0 for every x in the interval.
Note: The functions may be linearly independent even though W ( y1, y2 ,..., yn )  0
Definition 4.1.3 (Fundamental Set of Solutions)
Any set y1, y2 ,..., yn of n linearly independent solutions of the homogeneous linear nth-order
DE on an interval I is said to be a fundamental set of solutions on the interval.
Theorem 4.1.5 (General Solution of Homogeneous Equations)
Let y1, y2 ,..., yn be a fundamental set of solutions of the homogeneous linear nth-order
differential equation on an interval I. Then the general solution of the equation on the interval is
y  c1 y1  c2 y2  ...  cn yn
where ci , i  1, 2,..., n are arbitrary constants
Example: Verify that the given functions form a fundamental set of solutions of the DE on the
indicated interval. Form the general solution.
1)
y '' y ' 12 y  0; e3x , e4 x , (, )
2)
x2 y '' xy ' y  0; cos(ln x),sin(ln x), (0, )
3)
y '' 2 y ' 5 y  0; e x cos 2 x, e x sin 2 x , (, )