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6.5 Analytical Solutions of Constant Coefficients ODE
Finding the solutions of an ODE with constant coefficients is conceptually
very similar to solving the linear difference equation with constant coeffi-
cients. We repeat the exercise here for its pedagogical benefits and to bring
out some of the finer technical details peculiar to the ODEs of particular inter-
est for later discussions.
The linear differential equation of interest is given by:

n
dy d n−1 y dy
a + a +…+ a + ay = u t() (6.44)
n n n−1 n−1 1 0
dt dt dt
In this section, we find the solutions of this ODE for the cases that u(t) = 0 and
u(t) = A cos(ωt).
The solutions for the first case are referred to as the homogeneous solu-
tions. By substitution, it is a trivial matter to verify that if y (t) and y (t) are
2
1
solutions, then c y (t) + c y (t), where c and c are constants, is also a solution.
1
2
1 1
2 2
This is, as previously mentioned, referred to as the superposition principle
for linear systems.
If u(t) ≠ 0, the general solution of the ODE will be the sum of the corre-
sponding homogeneous solution and the particular solution peculiar to the
specific details of u(t). Furthermore, by inspection, it is clear that if the source
can be decomposed into many components, then the particular solution can
be written as the sum of the particular solutions for the different components
and with the same weights as in the source. This property characterizes a lin-
ear system.

DEFINITION A system L is considered linear if:

L c ut( ( ) + c u t( ) +…+ c u t( )) = c L ut( ( )) + c L u t( ( )) +…+ c L u t( ( )) (6.45)
11 2 2 nn 1 1 2 2 n n

where the c’s are constants and the u’s are time-dependent source signals.

6.5.1 Transient Solutions
To obtain the homogeneous solutions, we set u(t) = 0. We guess that the solu-
tion to this homogeneous differential equation is y = exp(st). You may won-
der why we made this guess; the secret is in the property of the exponential
function, whose derivative is proportional to the function itself. That is:

d(exp( st))
= sexp( st) (6.46)
dt

© 2001 by CRC Press LLC