Math 308 - Differential Equations Fall 2003
Homework Assignment 2
Due Wednesday, October 29.
1. Using Maple for Second Order Equations. Work through the examples shown in the handout
“Solving Second Order Differential Equations”, and then use Maple to solve the following problems.
(You should review the earlier Maple handouts, too.)
(a) Consider the differential equation
y00 + 3y0+y=t2e−tsin(t).
i. (By hand, not with Maple.) Write down the correct form of the guess that you would use to
find a particular solution to this problem by using the method of undetermined coefficients.
Do not solve for the coefficients! Just write down the final guess–with all its undetermined
coefficients–that is sure to work. Remember that this means you must find yhto determine
if your guess will work.
ii. Use the dsolve command in Maple to find the general solution. Look at the solution carefully,
and indicate which terms in the solution belong to yhand which belong to Yp. (You can print
the Maple session and make notes on the print-out that you hand in.)
(b) Use Maple to solve the initial value problem
y00 +y0+ 8y=te−2tcos(t), y(0) = 1, y0(0) = −1,
and plot y(t) for −1< t < 8.
2. A Deflection Anemometer. Consider a spring-mass system as shown in Figure 3.8.10 on page 200
of the text (but, for the moment, we use τfor time and yfor the displacement). The mass is m, the
spring constant is k, the damping coefficient is γ, and y(τ) is the displacement of the mass at time
τ. Assume that wind blows horizontally, parallel to the direction of motion of the mass, and that the
force exerted on the mass by the wind is proportional to the wind speed v(τ). Then Newton’s Second
Law gives
my00(τ) + γ y0(τ) + ky(τ) = cv(τ),(1)
where cis the proportionality constant that relates the wind speed to the force.
(a) Show that by changing to the new function udefined by
y(τ) = c
ku(ω0τ),where ω0=rk
m,
and then changing the independent variable from τto a new variable t=ω0τ, we can rewrite the
differential equation (1) as
u00(t) + bu0(t) + u(t) = w(t) (2)
where
b=γ
√mk ,and w(t) = v(t/ω0) = v(τ).
(By choosing our coordinates in this way, we see that the problem is greatly simplified; there is
really just one parameter that is important, not four.) We will use (2) for the rest of this problem.
Hint. If y(τ) = c
ku(ω0τ), then
dy
dτ =d
dτ ³c
ku(ω0τ)´=cω0
ku0(ω0τ) = c
√mk u0(t).
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