Math 213
Name:
Midterm Exam
1. [7 points] Find the general solution to the following differential equation.
xy 0 = x4 + 3y
2. [5 points] Given that y 0 = 5 + 8xy2 and y(0) = 1, use Euler’s Method with step size 0.1 to estimate
the value of y(0.2).
3. [12 points] In 1974, Stephen Hawking discovered that black holes emit small amounts of radiation,
causing them to slowly lose mass over time. According to Hawking, the mass M of a black hole obeys
the differential equation
k
dM
,
= −
dt
M2
where k = 1.26 × 1023 kg3 /year.
(a) [5 pts] Find the general solution to the above equation.
(b) [7 pts] After a supernova, the remnant of a star collapses into a black hole with an initial mass
of 6.00 × 1031 kg. Based on your answer to part (a), how long will it take for this black hole to
evaporate completely?
4. [12 points] A large tank initially contains 30 L of water mixed with 6 kg salt. Pure water is added to
the tank at a rate of 3 L/min. The contents of the tank are kept well-mixed, and saltwater is pumped out
of the tank at a rate of 2 L/min. Find a formula for the total amount of salt in the tank after t minutes.
5. [14 points] A culture of bacteria is growing according to the logistic equation
dP
= 0.0005P (120 − P),
dt
where t is time in minutes, and P(t) is the population in thousands. Given that P(0) = 20, find a formula
for P(t).
6. [6 points] Find the values of x and y that satisfy the following equation.
#
"
#
" #
"
8 3
3 1 3
+
x y = 2
6 3
2
2 4
7. [7 points] For what values of k is the following matrix invertible?

1 2 4



1 3 5
0 k 2
8. [8 points] Find a 2 × 2 matrix X that satisfies the equation
AX B = C,
#
"
#
"
#
"
8 3
2 1
5 7
,B=
, and C =
.
where A =
1 2
2 1
1 0
9. [7 points] Find the inverse of the following matrix. You must show your work to receive full credit.

1 3 0



 2 7 1 
1 3 1

0
8


10. [10 points (5 pts each)] For the following questions, let A =  5

0
2
0
2
0
0
0
0
9
4
5
5
3
3
7
0
4

0
7


4 .

0
3
(a) Compute the determinant of A. You must show your work to receive full credit.
(b) Compute the top-left entry of A−1 . You must show your work to receive full credit.
11. [5 points] In the following linear system, use Cramer’s rule to find formulas for x and y in terms of p
and q.
px + 3y = q
x + qy = 2
12. [7 points] Find the determinant of the following matrix. You must show your work to receive full
credit.

1
4

 −2 −8

 1
4
1
7
2
1
2
8
4
2
7
5




