Feb 10 homework

For these first two, solve, graph, discuss limits, ... whatever can be done.
In each case, describe what method(s) worked.  
All of these are from Boyce/DiPrima 9th, chap 2.

t (dy/dt)+ 2 y = (t^2)-t+1
with y(t=1)=1/2, for t > 0.

x (dy/dx)=(Sqrt[1-(y^2)])

A ball with mass 0.15 kg is thrown upward with initial velocity 20 m/sec from the roof of a building 20 m high.

The equation of motion is
	(dv/dt)m = - m g - AirFriction
where the friction term is always in the opposite direction from the velocity.

For each of the following 3 cases, find (i) the maximum height, (ii) the time when the ball hits the ground (assuming it doesn't hit the building), and (iii) plot height vs time.  (All three on the same graph, please.)

a) Air friction = 0

b) Air friction = abs (v)/(m/sec) (1/30) Newtons

c) Air friction = (v^2)/(((m/sec))^2)(1/1325) Newtons

where a "Newton" is the unit of force in the MKS system,

Epidemic

Assume a given population can be divided into sick and healthy, with y the proportion of the sick.  Since people getting infected are proportional to both the number of sick people (y) and the number of healthy people (1-y), the rate of infection can be modeled as
	(dy/dt)=α y (1-y)

where α is a constant.  Let (Subscript[y, 0])be the initial proportion of sickies.

a) Find the equilibrium points for the system, and determine whether each is stable, unstable, or semistable.

b) Solve the initial value problem and verify that the conclusions you reached in a) are correct.  Show that y(t)->1as t-> ∞ in this model (i.e. everyone gets sick).

Euler