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
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Practice Problems: Midterm Exam
1. Find the general solution to each of the following differential equations.
(a) y ′′ = 1+ e2x
(b) y ′ = x + xy2
(c) xy ′+3y = x2
(d) y ′ + (xy)2 = 0
(e) y ′ = ex − 2y
(f) ln(y ′) + 2ln(x) = y
2. Given that y ′ = 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. Find the solution to the following initial value problem.
2x2y ′′ − 3xy ′ − 3y = 0, y(2) = 8, y ′(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:
2NO + Cl2 −→ 2NOCl.
The rate of this reaction is governed by the equation
d[NO]
dt= −k [NO]2
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 concen-
tration has decreased to 0.122 mM. What is the value of the constant k?
5. Find all values of r for which y = erx is a solution to the following equation:
y ′′ + 2y ′ − 15y = 0.
6. Find a 2×2 matrix X that satisfies the equation
AX = BX + CT ,
where A =
[5 53 3
], B =
[1 30 1
], and C =
[1 23 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.
(a)
[7 35 4
]
(b)
1 3 22 7 33 7 9
11. Compute the following determinants.
(a)
∣∣∣∣∣∣∣1 2 06 8 34 2 1
∣∣∣∣∣∣∣
(b)
∣∣∣∣∣∣∣∣∣∣∣
1 3 2 2 72 1 3 0 60 0 3 0 00 0 9 0 54 7 6 0 2
∣∣∣∣∣∣∣∣∣∣∣
(c)
∣∣∣∣∣∣∣∣∣∣∣
1 2 3 2 11 7 4 6 12 4 8 7 51 2 3 2 53 6 9 8 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.
13. Let A =
3 4 2 91 2 0 32 3 4 82 3 4 9
. Given that det(A) = 6, compute the top-left entry of A−1.
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