162 Ordinary Differential Equations
is k = 5 x 10-^12 /((° K)^3 -min), and which is surrounded by a mediumwith constant temperature A = 500° K. (° K is degrees Kelvin. On
the Kelvin temperature scale 0° corresponds to absolute zero and one
Kelvin degree is the same size as one Celsius degree.)- When a periodic voltage E(t) = Eo sin wt is suddenly applied to a coil
wound around an iron core, the m agnetic flux, y(t), satisfies the initial
value problem
y' = -Ay-By^3 + Eosinwt; y(O) = 0
where A, B, E 0 , and w are positive constants. Solve this initial value
problem and graph the solution on the interval [-.5, .5] for A= 2, B = 3,Eo = 2, and w = 4.
- A simple model for the spread of a rumor assumes there is a fixed
population size, N, and that each person who has heard the rumor tells
the rumor to m people each day. Some of them will already have heard
the rumor. Let H(t) b e the number of people who have heard the rumor
at time t. Each day each person who has heard t he rumor will tell it
to m(N - H(t))/N people who have not already heard the rumor, so
H(t) satisfies the differential equation H'(t) = H(t)m(N - H(t))/N.
Assume N = 500, m = 3, and H(O) = 1. Compute and graph H(t) on
the interval [O, 10]. When is H(t) = 250? What is limt--+oo H(t)?
- Compute and graph on the interval [l , 4] the curve which lies above the
x-axis and has the property that the length of the a rc joining any two
points is equal to the area under the arc. - In 1913 , L. Michaelis and M. Menton developed the following simple
model for chemical enzyme kinetics dy / dt = -y / (y + 1); y( 0) = 1.
Solve and graph the solution to this init ial value problem on the interval
[O, 5].