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In many physical problems of interest, we desire solutions that are zero at
infinity, that is, decay over a finite time. This requires that both α and α be
1
2
negative; or if only one of them is negative, that the c coefficient of the expo-
nentially increasing solution be zero. This class of solutions is called the over-
damped class.
• If b – 4ac = 0, the two roots are equal, and we call this root α degen. .
2
The solution to the differential equation is

y = c ( + c t)exp(α t) (6.52)
homog. 1 2 degen.

The polynomial, multiplying the exponential function, is of degree one here
because the degeneracy of the root is of degree two. This class of solutions is
referred to as the critically damped class.

2
• If b – 4ac < 0, the two roots are complex conjugates of each other,
and their real part is negative for physically interesting cases. If we
denote these roots by s = –α ± jβ, the solutions to the homogeneous
±
differential equations take the form:
y homog. = exp(–αt)(c cos(βt) + c sin(βt)) (6.53)
1
2
This class of solutions is referred to as the under-damped class.

In-Class Exercises

Find and plot the transient solutions to the following homogeneous equa-
tions, using the indicated initial conditions:

Pb. 6.28 a = 1, b = 3, c = 2 y(t = 0) = 1 y′(t = 0) = –3/2
Pb. 6.29 a = 1, b = 2, c = 1 y(t = 0) = 1 y′(t = 0) = 2
Pb. 6.30 a = 1, b = 5, c = 6 y(t = 0) = 1 y′(t = 0) = 0

6.5.2 Steady-State Solutions

In this subsection, we find the particular solutions of the ODEs when the
driving force is a single-term sinusoidal.
As pointed out previously, because of the superposition principle, it is also
possible to write the steady-state solution for any combination of such inputs.
This, combined with the Fourier series techniques (briefly discussed in Chap-
ter 7), will also allow you to write the solution for any periodic function.

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