**MA2051--C97****SOLUTIONS FOR TEST I**
**1.**
You are solving

**(a)**-
The corresponding homogeneous equation is
.
The characteristic equation is
*r*+3 = 0, so*r*=-3 and the homogeneous solution is . **(b)**-
Use the method of Undetermined Coefficients and look for a solution of
the form
. Plug this into the original
equation and equate coefficients to obtain
*A*= 7, so . **(c)**-
The general solution is .
The initial condition gives you 5 =
*y*(0) =*C*+ 7 and so*C*= -2. **(d)**-
The answer is yes: simply choose
(and remember that solutions are unique).

**(a)**-
Sugar enters the tank at rate
(Tbsp/min).
**(b)**-
Sugar leaves the tank at the rate
(Tbsp/minute).
The assumption that the tank is ``well-mixed'' implies that the
concentration of sugar is uniform throughout the tank.
**(c)**-
The balance equation is
Net change between
*t*and= (rate in) (rate out) ,

or

Collect terms, divide by and take the limit as approaches zero to obtain the model:

**(d)**-
The equation has a steady state at , so the
concentration will be close to
*C*= 22/3 = 7.333 if you wait long enough.

gives you

when you plug in the initial data.

The disaster occurs as *t* approaches 5/*k*, when the population
becomes larger than the number of electrons in the universe.

**4.**
The differential equation tells you that
The steady states are *y*=+6 and *y*=+9, where .

when (or where) *y*(*t*) < 6 or *y*(*t*)> 9

when 6 < *y*(*t*) < 9

Your graph starting at *y*=0 should increase, be concave down, and
have a horizontal asymptote at *y*=6.
The graph starting at *y*=12 should be increasing and concave up.
The graph starting at *y*=8.9 should decrease, be concave down
initially but switch to concave up (at *y*=7.5), and
have a horizontal asymptote at *y*=6.
**Bonus Problem**
Define *Q*(*t*) = 1/*P*(*t*) and differentiate it:

but this simplifies to

This is linear for *Q* and the solution is

Thu Feb 6 17:34:47 EST 1997