Math 105/206 - Quiz 4, Mar 13 2015
IMPORTANT: Write your name AND student number somewhere on this sheet.
No calculators, books or notes. Please show your work to get full marks. (10 marks total)
Problem 1 (4 marks)
A population P (t) of mice grows proportionally to the square root
3. If P (0) = 100, determine P (t).
p
P (t), with a constant of proportionality
Solution
We have to solve the first-order initial value problem
√
0
P =3 P
P (0) = 100
Separating the variables in the equation we get
dP
√ = 3dt
P
and integrating
Z
Now
so
dP
√ =
P
Z
Z
3dt = 3t + C.
√
dP
√ = 2 P + C0
P
√
2 P = 3t + C
(we can ignore one of the arbitrary constants, because it gets ’incorporated’ in the other one).
Solving for P we get
(3t + C)2
P (t) =
4
and imposing the initial condition
C2
100 = P (0) =
4
gives C = 20 or C
√ = −20. To choose between the two it is sufficient to notice that by one of the equations
above 3t + C = 2 P ≥ 0, so C has to be ≥ 0 (set t = 0, for example). Alternatively, using the same equation
you can write directly
p
20 = 2 P (0) = 3 · 0 + C = C.
So our solution is
P (t) =
(3t + 20)2
.
4
Problem 2
Solve the following first-order separable differential equation
x3 y 0 = y 2 + 1
Solution
Separating the variables we get
dy
= x−3 dx
+1
y2
and integrating
1
arctan(y) = − x−2 + C.
2
Solving for y gives
1
y(x) = tan − 2 + C .
2x
Problem 3
Determine whether the following functions are CDFs of
answer, don’t just write ’yes’ or ’no’.

 0
(x + 1)2
F1 (x) =

1
a continuous random variable. Please justify your
if x ≤ −1
if − 1 < x ≤ 0
if x > 0
1
F2 (x) = arctan −
x
Solution
The function F1 is a CDF: it is continuous since (x + 1)2 at x = −1 gives 0 and at x = 0 gives 1, it is
non-decreasing since (x + 1)2 is (its derivative is 2(x + 1), which is non-negative for x ≥ −1), and the limits
check out, because of how the function is defined (it is constantly 0 for x ’very negative’ and constantly 1 for x
’very positive’).
The function F2 is not a CDF because it is not continuous: x1 is not defined in x = 0, and
1
π
lim− arctan −
=
x→0
x
2
(since − x1 goes to +∞ if x approaches 0 from the left)
1
π
lim+ arctan −
=−
x→0
x
2
(since − x1 goes to −∞ if x approaches 0 from the right).