Practice Problems: Midterm Exam
1. Find the general solution to each of the following differential equations.
(a) y 00 = 1 + e 2 x
(b) y 0 = x + xy 2
(c) xy 0 + 3y = x 2
(d) y 0 + (xy)2 = 0
(e) y 0 = ex − 2y
(f) ln(y 0 ) + 2 ln(x) = y
2. Given that y 0 = 8x 2 + 10y 2 and y(0) = 0.2, use Euler’s method with step size 0.25 to estimate
the value of y(0.5).
3. Find the solution to the following initial value problem.
2x 2 y 00 − 3xy 0 − 3y = 0,
y(2) = 8,
y 0 (2) = 12.
4. During a chemistry experiment, a small amount of gaseous nitric oxide (NO) is added to a
sample of chlorine gas (Cl2 ). The nitric oxide reacts with the chlorine, producing nitrosyl
chloride:
2 NO + Cl2 −→ 2 NOCl.
The rate of this reaction is governed by the equation
d[NO]
= −k [NO]2
dt
where [NO] is concentration of nitric oxide in millimolars (mM), and t is the time in hours.
(a) Find the general solution to this differential equation.
(b) Suppose the initial concentration of nitric oxide is 0.200 mM. After one hour, the concentration has decreased to 0.122 mM. What is the value of the constant k?
5. Find all values of r for which y = e rx is a solution to the following equation:
y 00 + 2y 0 − 15y = 0.
6. Find a 2 × 2 matrix X that satisfies the equation
AX = BX + C T ,
5
5
1
3
1
2
where A =
,B=
, and C =
.
3 3
0 1
3 4
7. Find a 2 × 2 diagonal matrix A such that tr(A) = 11 and det(A) = 28.
8. Use one or more parameters to describe the solution set of the following linear system.
x1 + 3x2 + 2x3 + x4 = 4
2x1 + 6x2 + 7x3 + 8x4 = 14
x1 + 3x2 + 4x3 + 5x4 = 8
9. 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?
10. Find the inverses of the following matrices.
#
"
7 3
(a)
5 4


1 3 2


(b)  2 7 3 
3 7 9
11. Compute the following determinants.
1 2 0
(a) 6 8 3 4 2 1
1
2
(b) 0
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
2
6
7
2
8
1 1 5
5 9
12. 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
13. Let A = 
2
2
4
2
3
3
2
0
4
4

9

3
. Given that det(A) = 6, compute the top-left entry of A−1 .
8
9