MATH 282 - DE
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4.2 Sinusoidal Forcing
Today we are going to study forced equations where the forcing function is sinusoidal
(either sine or cosine). Let’s apply the guessing technique to sinusoidally forced linear
equations:
Example. Let’s calculate the general solution to the equation:
𝑑2 𝑦 𝑑𝑦
+
+ 2𝑦 = cos 2𝑡
𝑑𝑡 2 𝑑𝑡
Solution:
1
Here are the graphs of three solutions:
Here is the graph of the steady-state solution:
A little translation:
Consider the 2nd order linear forced equation
𝑑2𝑦
𝑑𝑦
𝑚 2 +𝑏
+ 𝑘 𝑦 = 𝑓(𝑡)
𝑑𝑡
𝑑𝑡
where 𝑚 and 𝑘 are positive and 𝑏 ≥ 0.
Engineering Terminology:

forced response – any solution to the forced equation.

steady-state response – behavior of the forced response over the long term.

natural (or free) response – any solution of the associated homogeneous equation.
Why are initial conditions essentially irrelevant?
When we guess a solution of the form 𝑦𝑐 (𝑡) = 𝛼𝑒 𝑖𝜔𝑡 and compute the complex number 𝛼, we
have essentially determined everything we need to know about the steady-state solution.
Euler’s formula gives us a nice way of determining the amplitude, and phase-angle of the
steady-state solution immediately from 𝛼.
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Why we prefer to calculate the steady-state solution using complex numbers?
Recall that we calculated the steady-state solution
1
𝑦𝑝 (𝑡) = − (cos 2𝑡 − sin 2𝑡)
4
for the equation
𝑑 2 𝑦 𝑑𝑦
+
+ 2𝑦 = cos 2𝑡,
𝑑𝑡 2 𝑑𝑡
and we did so using computations that involved complex numbers. In fact, we found 𝑦𝑝 (𝑡) as
the real part of
1
𝑦𝑐 (𝑡) = − (1 + 𝑖)𝑒 (2𝑖)𝑡 .
4
The complex number
1
𝛼 = − (1 + 𝑖)
4
Tells us everything we need to know about the steady-state solution.
In order to see why, we use polar coordinates in the complex plane ( see pp. 745-747 in
Appendix C of the text).
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