Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
WorkSheet 10
I) Solve the following di¤erential equations by hand.
(a)
dy
x
=
dx
y
(b)
y
dy
= ;
dt
t
(c)
dx
= 2x
dt
(d)
dy
= 3y;
dt
(e)
dy
= ¡2y + 6
dx
(f)
dx
= ¡2x + 6 ;
dt
y(1) = 1
y(0) = 4
x(0) = 9
II) Set us a di¤erential equation for each of the following situations. You need not solve the
DE at this time.
(a) Assume that the rate of growth of a bacteria is proportional to the amount present. If
P (0) = 3g and P (1) = 3:2g; then …nd the amount at any time t:
(b) Suppose that you deposit $2000 into an account that pays 7.5% interest compounded
continuously. How much will be in the account 3 years later?
(c) Assume that a cup of hot water (200 F) is put into a room (78 F) at noon on Friday. If
the temperature at 12:15 is 120 F, …nd the temperature at any time t: Use Newton’s Law of
Cooling.
(d) A 100 gallon tank initially has a 100 gallon brine solution containing 20 lb of salt. If
pure water enters the tank at a rate of 2 gal/min and the mixture leaves the tank at the
same rate, …nd the amount of salt in the tank at any time t:
(e) Same as (d) except assume that the water entering the tank contains 1/4 lb per gallon.
Now …nd the amount of salt in the tank at any time t: