Practice Problems: Midterm Exam
1. Find the solution to each of the following initial value problems.
(a) y 0 + (xy)2 = 0,
y(0) = 1
(b)
dy
= ex − 2y,
dx
y(0) = 2
y(0) = 3
dy
= x + xy2 ,
dx
y(0) = 1
(d) y 0 = y(2 − y),
(e) xy 0 − 3y = x2 ,
y(1) = 3
(f) ln(y0 ) + 2 ln(x) = y,
(c)
y(1) = ln(2)
2. Given that y 0 = 8x2 + 10y2 and y(0) = 0.2, use Euler’s method with step size 0.25 to estimate
the value of y(0.5).
3. A cup of water with an initial temperature of 70◦ C is left on a counter to cool. After ten minutes,
the temperature has decreased to 60◦ C. Assuming the air temperature is 20◦ C, use Newton’s
Law of Cooling to predict the temperature of the water after one hour.
4. A chemist prepares a sample of
210 Po.
Because of radioactive decay, the mass of the sample
decreases by 3.5% each week. What is the half life of 210 Po?
5. A ball is dropped from the top of a tall building. If air resistance is taken into account, the
downward velocity v of the ball obeys the differential equation
dv
= 9.8 1 − 0.01v2 ,
dt
where t is measured in seconds and v is measured in meters/second. Assuming the initial
velocity of the ball is zero, find a formula for the velocity of the ball after t seconds.
6. Find a 2 × 2 matrix X that satisfies the equation
AX =
5
5
1
3
where A =
,B=
, and C = 1
3 3
0 1
3
BX + C,
2 .
4
7. Consider the following system of equations involving the variables x, y, and z:
x + 3y + 2z = 4
2x + 7y + 6z = 10
y + kz = 3
(a) For what values of k does this system have a unique solution?
(b) For what values of k does this system have infinitely many solutions?
(c) For what values of k is this system inconsistent?
8. Find the values of x and y that satisfy the following matrix equation:


1 0

3 x 0 
y
0
 2 0 =
y 0 2
x 4
−2 2
9. Find the inverses of the following matrices:


"
#
1 3 2
7 3


(b)  2 7 3 
(a)
5 4
3 7 9
10. Compute the following determinants:
1
1 2 0
2
(b) 0
(a) 6 8 3 4 2 1
0
4
3
1
0
0
7
2
3
3
9
6
2
0
0
0
0
7 6 0
5 2
1
1
(c) 2
1
3
2
7
4
2
6
3
4
8
3
9
11. Consider the following linear system:
x + ky + 2z = 0
2x + 7y + kz = 2
x + 4y + 4z = 3
Use Cramer’s rule to find a formula for y in terms of k.

3

1
12. Let A = 
2
2
4
2
3
3
2
0
4
4

9

3
. Given that |A| = 6, compute the top-left entry of A−1 .
8
9
2
6
7
2
8
1 1 5
5 9