Covered today: Homogeneous 2nd order constant coefficient DEs for which the characteristic equation has complex conjugate solutions. We now know (but still haven't proven) the solution to any homogeneous 2nd order constant coefficient DE. We'll prove that we actually have the general solution next week.

For next time: 3.4 - 7,11,17,22 and 3.8 - 5,9 (for these last two, simply set up and solve the DE - don't worry about the quasi-frequency and the quasi-period and whatnot (yet)). Also, do the following problem: assume you have a spring-mass system which is critically damped. Suppose the mass starts at equilibrium and you give it an initial velocity (in either direction). Prove that the mass will never return to its equilibrium.

Transparencies from class: (these were taken almost verbatim from the more complete discussion in section 3.8 of your book)

Consider a spring (which we'll assume to be weightless) whose natural length is l. Suppose that we hang a mass m at the end of the spring, and this elongates the spring by L units. There are now two forces acting on the mass: its weight, which is acting downward, and a spring force F_s which is acting upward. Hooke's law states that (for reasonably small values of L, which we'll assume we have at all times) the spring force is proportional to the elongation of the spring. The constant of proportionality k is called the spring constant.

Assuming downward is the positive direction, explain why mg-kL=0.

We will now consider what happens to the position of this mass at the end of this spring if it is initially displaced, given an initial velocity, and/or acted on by an outside force. Let u(t) denote the position of the mass at time t, with u=0 corresponding to the equilibrium position of the mass (i.e. when u=0 the spring is L+l units long), and with u(t) being measured positive downward.

There are four forces acting on the mass at any time t.

1.

the weight of the mass;

2.

the spring force F_s(t), which is proportional to the total elongation L+u(t) of the spring (with constant of proportionality equal to the spring constant k) and always acts to restore the spring to its equilibrium position. Show that F_s(t)=-k(L+u(t)). Why is the sign in front always the same despite the fact that this force is acting in different directions at different times?

3.

the resistive force (or damping force) F_d(t), which is proportional to the speed of the mass. Show that $F_d(t)=-\gamma u'(t)$, where $\gamma$ is a positive constant called the damping constant. Again, you should convince yourself that this one formula is valid at all times whether the mass is moving upward or downward.

4.

an applied external force F_e(t).

Prove that, if there is no external force, this system is described by a homogeneous 2nd-order linear constant-coefficient DE (with nonnegative coefficients).

Suppose a mass weighing 5 pounds is hooked to a spring and stretches it 2 inches. Assuming no air resistance, if this mass is pushed up 1 foot and then released, find the function u describing the position of the mass at time $t \geq 0$.

Suppose the same mass is suspended from the same spring, but now we assume that there is air resistance given by $\gamma = 3$. Find the function u describing the postion of the mass at time $t \geq 0$.