HOMOGENEOUS LINEAR SYSTEM WITH CONSTANT COEFFICENTS:

1
HOMOGENEOUS LINEAR SYSTEM WITH CONSTANT
COEFFICENTS TWO EQUATIONS IN TWO UNKNOWN
FUNCTIONS
2
We Shall concerned with the homogeneous linear
system
where the coefficients a1, a2, b1, and b2 are
real constants. We seek solution of this system
but how shall we proceed?
3
If we recall that the exponential function has
the property that its derivatives are constant
multiples of the function itself then, let us
therefore attempt to determine a solution of the
form
where A, B, and m are constant. If we substitute
(2) into (1), we obtain
4
These equations lead at once to the system
in the unknowns A and B. This system obviously
has the trivial solution A B 0. But this
would only lead to the trivial solution x y
0 of system (1).
5
Thus we seek nontrivial solutions of the system
(1). A necessary and sufficient condition that
this system have a nontrivial solution is that
the determinant
Expanding this determinant we led at once to the
quadratic equation
6
in the unknown m. This equation is called the
Characteristic equation associated with the
system (1). Its roots m1 and m2 are called the
Characteristic roots.
Suppose m m1. Then substituting m m1 into
the algebraic system (3), we may obtain a
nontrivial solution A1, B1 of this algebraic
system. With these values A1, B1 we obtain the
nontrivial solution
7
of the given system (1)

• Three cases must now be considered
• The roots m1 and m2 are real and distinct.
• The roots m1 and m2 are conjugate complex.
• The roots m1 and m2 are real and equal.

8
Case I The roots of the Characteristic equations
(5) are real and distinct
Hypothesis
The roots m1 and m2 of the Characteristic
equation (5) associated with the system (1) are
real and distinct.
9
Conclusion The system (1) has two nontrivial
linearly independent solutions of the form
and
where A1, A2, B1, and B2 are definite constants.
The general solution of the system (1) may thus
be written
10
where c1 and c2 are arbitrary constants.
11
Example
Solution
We assume a solution of the form
12
Substituting (7) into (6) we obtain
and this leads at once to the algebraic system
In the unknown m. For nontrivial solutions of
this system we must have
13
Expanding this we obtain the characteristic
equation
Solving this we find the roots m1 3, m2 4.
14
Setting m m1 3 in (8), we obtain
A simple nontrivial solution of this system is
obviously A B 1. With these values of A, B,
and m we find the nontrivial solution
15
Now setting m m2 4 in (8), we find that
A simple nontrivial solution of this system is
obviously A 3, B 2. With these values of A,
B, and m we find the nontrivial solution
16
The solutions (9) and (10) are linearly
independent, and the general solution of the
system (6) may be written
17
Case II. The roots of the Characteristic equation
are conjugate complex
If m1 and m2 of the charactrisic equation (5)
are the conjugate complex numbers then
we still obtain two distinct solutions

and
18
Theorem Hypothesis
The roots m1 and m2 of the characteristic
equation (5) associated the system (1) are the
conjugate complex numbers
Conclusion The system (1) has two real linearly
independent solutions of the form
19
where A1, A2, B1, and B2 are definite real
constants. The general solution of the system (1)
may thus be written
20
where c1 and c2 are arbitrary constants.
21
Example
We assume a solution of the form
Substituting (12) into (11) we obtain
22
and this leads at once to the algebric system
in the unknown m. For nontrivial solutions of
this system we must have
23
Expanding this, we obtain the characteristic
equation
The roots of this equation are the conjugate
complex numbers
24
Setting m 23i in (13), we obtain
A simple nontrivial solution of this system is A
2, B -13i. Using these values we obtain the
complex solution
25
Using Eulers formula this takes the form
Since both the real and imaginary parts of this
solution of system (11) are themselves solution
of (11), we thus obtain the two real solutions
26
and
27
Finally, since the two solutions are linearly
independent we may write the general solution of
the system (11) in the form
Where c1 and c2 are arbitrary constants.
28
Case III. The roots of the Characteristic
equation are real and equal
If m1 and m2 of the charactrisic equation (1)
are real and equal, then we still obtain two
distinct solutions
and
29
Theorem Hypothesis
The roots m1 and m2 of the characteristic
equation (5) associated the system (1) are real
and equal. Let m denote their common value.
Conclusion The system (1) has two linearly
independent solutions of the form
30
where A, A1, A2, B, B1, and B2 are definite real
constants, A1 and B1 are not both zero and B1/A1
B/A. The general solution of the system (1) may
thus be written
31
where c1 and c2 are arbitrary constants.
32
Example
We assume a solution of the form
Substituting (16) into (15) we obtain
33
and this leads at once to the algebric system
in the unknown m. For nontrivial solutions of
this system we must have
34
Expanding this we obtain the characteristic
equation
Thus the characteristic equation (18) has the
real and equal roots 3, 3.
35
Setting m 3 in (18), we obtain
A simple nontrivial solution of this system being
A B 1, we obtain the nontrivial solution
of the given system (15)
36
Since the roots of the characteristic equation
are both equla to 3, we must seek a second
solution of the form (14) with m 3. That is,
we must determine A1, A2, B1, and B2 ( with A1
and B1 not both zero) such that
37
is a solution of the system (15). Substitutiong
(20) into (15), we obtain
These equations reduce at once to
38
In order for these equations to be identities, we
must have
39
Thus in order for (21) to be a solution of the
system (15), the constsnts A1, A2, B1, and B2
must be chosen to satisfy the equation (). From
the equations A1 B1 0, we see that A1
B1. The other equations of (21) show that A2 and
B2 must satisfy
40
We may choose any convenient nonzero values for
A1 and B1. We Choose A1 B1 1. Then (22)
reduces to A2 B2 1, and we can choose any
convenient values for A2 and B2 that will
satisfy this equation. We choose A2 1, B2 0.
We are thus led to the solution
41
By Theorem the solutions (19) and (23) are
linearly independent. We may thus write the
general solution of the system (15) of the form
Where c1 and c2 are arbitrary constants.
42
We note that a different choice of nonzero values
for A1 and B1 in (22) and/or a different choice
for A2 and B2 in the resulting equation for A2
and B2 will led to a second solution which is
different from solution (23). But this different
second solution will also be linearly
independent of the basic solution (19), and hence
could serve along with (19) as one of the two
constitute parts of a general solution.
43
For example, if we choose A1 B1 2 in (22),
then (22) reduced to A2 B2 2 and if we then
chose A2 3, B2 1, we are led to the different
second solution
This is linearly independent of solution (19) and
could seve along with (19) as one of the two
constitutent parts of a general solution.
44
Method of Variation of Parameter (Finding
particular solution of nonhomogeneous linear
systems)
Consider the nonhomogeneous linear system
45
and the corresponding homogeneous system
If
and
46
are linearly independent solution of (2), so that
its general solution is
47
will be a particular solution of (1) if the
functions v1(t) and v2(t) satisfy the system
48
Example Find the general solution of the system
49
Solution Corresponding homogeneous system is
its general solution is
50
Formula for particular solution is
51
Solving these two equations (for v1 and v2)
52
Particular Solution is