MATH 2260 - 41514 QUIZ 4 MODEL SOLUTION
(1) (5 pts) Solve the differential equation
dy
ey x
= 1.
dx
1
ey dy = dx (2 pts)
x Z
Z
1
ey dy =
dx
x
Z
ey dy = ey + C1 (3 pts)
Z
1
dx = ln |x| + C2 (4 pts)
x
ey + C1 = ln |x| + C2
ey = ln |x| + C (5 pts)
(2) (5 pts) A colony of bacteria is grown under ideal conditions in a laboratory
so that the population increases exponentially with time. After 5 hours, the
number of bacteria is doubled. How long does it takes to be 3 times of the
initial number?
y(t) = number of bacteria after t hours,
y(t) = y0 ekt (1 pt)
y0 = y(0)
2y0 = y(5) = y0 e5k ⇒ e5k = 2 ⇒ 5k = ln 2 ⇒ k =
1
ln 2
5
1
y(t) = y0 e( 5 ln 2)t (3 pts)
1
1
y(t) = 3y0 ⇒ 3y0 = y0 e( 5 ln 2)t ⇒ e( 5 ln 2)t = 3
1
5 ln 3
⇒
ln 2 t = ln 3 ⇒ t =
(4 pts)
5
ln 2
5 ln 3
t=
hours (5 pts)
ln 2
Date: February 14, 2013.
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