Math 215
Spring 2010
Assignment 4
You are highly recommended to own a textbook, Boyce-DiPrima or another, to see a comprehensive treatment of the subjects and more examples.
§3.6: 8, 11, 13, 15, §3.7: 1, 7, 9, 20
• (§3.6: 8) Find the general solution of y 00 + 4y = 3 csc 2t,
0 < t < π/2.
• (§3.6: 11) Find the general solution of y 00 − 5y 0 + 6y = g(t). Here g(t) is an arbitrary
continuous function.
• (§3.6: 13) Verify that the given functions y1 and y2 satisfy the corresponding homogeneous equation; then find a particular solution of the given nonhomogeneous
equation.
t2 y 00 − 2y = 3t2 − 1,
t > 0;
y1 (t) = t2 ,
y2 (t) = t−1 .
• (§3.6: 15) Verify that the given functions y1 and y2 satisfy the corresponding homogeneous equation; then find a particular solution of the given nonhomogeneous
equation.
ty 00 − (1 + t)y 0 + y = t2 e2t ,
t > 0;
y1 (t) = 1 + t,
y2 (t) = et .
• (§3.7: 1) Determine ω0 , R and δ so as to write the expression u = 3 cos 2t + 4 sin 2t in
the form u = R cos(ω0 t − δ).
• (§3.7: 7) A mass weighing 3 lb stretches a spring 3 in. If the mass is pushed upward,
contracting the spring a distance of 1 in., and then set in motion with a downward
velocity of 2 ft/s, and if there is no damping, find the position u of the mass at any
time t. Determine the frequency, period, amplitude, and phase of the motion.
• (§3.7: 9) A mass of 20 g stretches a spring 5 cm. Suppose that the mass is also
attached to a viscous damper with a damping constant of 400 dyn·s/cm. If the mass
is pulled down an additional 2 cm and then released, find its position u at any time t.
Plot u versus t. Determine the quasi frequency and the quasi period. Determine the
ratio of the quasi period to the period of the corresponding undamped motion. Also
find the time τ such that |u(t)| < 0.05 cm for all t > τ .
• (§3.7: 20) Assume that the system described by the equation mu00 + γu0 + ku = 0
is critically damped and that the initial conditions are u(0) = u0 and u0 (0) = v0 . If
v0 = 0, show that u → 0 as t → ∞ but that u is never zero. If u0 is positive, determine
a condition on v0 that will ensure that the mass passes through its equilibrium position
after it is released.