Math 2320 Review Items for the Final Exam
Final Exam Dates: Friday, December 10, Saturday, December 11, or
Sunday, December 12
You may bring your calculator (no TI 89’s) and one page of handwritten notes. These items consist of assigned
homework items. You should be able to find the worked out solutions within the Homework Solutions pages.
Section
Objective
Items
/Page
1.1
Give the order and determine if 1. 1 − " − 4′ + 5 = cos p.10 the equation is linear or
Solution: 2nd order, linear
nonlinear
3. − " + 6 = 0
Solution: 4thd order, linear
5.
= 1 + Solution: 2nd order, nonlinear
7. sin ! """ − cos ! " = 2
Solution: 3rd order, linear
2.1
p.48
Find critical points and sketch
an approximate solution curve
passing through a point.
= − 3
21.
Solution: y = 0 stable, y=3 unstable
23.
= − 2
Solution: Critical Points y = 2 semi−stable
25. = 4 − Solution: y = 0 semi−stable, y=2 stable,
y = −2 unstable
Ch. 2
Classify and solve 1st order
ODEs
p. 54 #17
%
&
= ' − '
Solution: ' =
() *
+,() *
p. 65 #13 " + + 2 = - Solution: =
).
+
/) 0.
p. 73 #13 = 2- − + 6 Solution: − 2 − 1- − 2 + 1 = 0
p. 78 #5 + 2 − 2 = 0
3
Solution: =
p.78 #17
= − 1
3
Solution: 45||,/
+
= + + 1- 3.1 p.
98
Linear Models
3. The population of a town grows at a rate proportional to the
population present at time t. The initial population of 500 increases
by 15% in 10 years. What will be the population in 30 years? How
fast is the population growing at t = 30?
Solution:
15. A small metal bar, whose initial temperature was 20°C, is
dropped into a large container of boiling water. How long will it
take the bar to reach 90°C if it is known that its temperature
increases 2° in 1 second? How long will it take the bar to reach
98°C?
Solution:
21. A large tank is filled to capacity with 500 gallons of pure water.
Brine containing 2 pounds of salt per gallon is pumped into the
tank at a rate of 5 gal/min. The well−mixed solution is pumped out
at the same rate. Find the number A(t) of pounds of salt in the tank
at time t.
Solution:
4.3 p.
147
Homogeneous Linear Equations Solve each equation.
with Constant Coefficients
1. 4" + ′ = 0
Solution:
9. " + 9 = 0
Solution:
37. " − 10′ + 25 = 0, 0 = 1,
1 = 0
Solution:
4.4
p.158
Undetermined Coefficients
13. Solve " + 4 = 3 sin2
Solution:
6.1 p.
248
Series Solutions of Linear
Equations
11. Rewrite the given expression as a single power series whose
general term involves ( .
;
;
5<+
5<=
9 2:15 53+ + 9 615 Solution:
5,+
17. Find a power series solution to the following ODE. Let
∞
y=
∑c x
n
n
n =0
Solution:
.
" − = 0
7.1 p.
283
Definition of the Laplace
Transform
11. Use the definition to find ℒ?@A, where @ = - &,B
Solution:
7.2 #
292
+
Use appropriate algebra and
29. ℒ 3+ CD
E
, +D , Theorem 7.4 to find the inverse
Solution:
Laplace transform.
7.2 p.
292
Solve an ODE by applying
Laplace transforms.
33. " + 6 = - & , 0 = 2
Solution:
37. " + = √2sin√2, 0 = 10, ′0 = 0
Solution:
7.3 p.
301
Translation on the s−Axis
+
13. ℒ 3+ C E
D 3 GD,+=
Solution:
7.3 p.
302
Translation on the t−Axis
47. ℒ 3+ C
Solution:
) 0H
DD, +
E
, 0 ≤ < 2L
59. ℒ 3+ ?@A, @ = C
0,
≥2
Solution:
8.2 p.
351
Find the general solution of the
given system.
3.
&
= −4 + 2
= − + 2
Solution:
&
10 −5
5. M " = M
8 −12
Solution:
9.1 p.
372
Use the improved Euler's
1. " = 2 − 3 + 1, 1 = 5; 1.5
method to obtain a
Solution:
four−decimal approximation of
x(n) y(n)
the indicated value. Use h = 0.1. 1.0 5.0000
Also find the percent error.
1.1 3.9900
1.2 3.2546
1.3 2.7236
1.4 2.3451
1.5 2.0801
+
R
Solution to ODE: = 6 + 1 + - 3
Q
Q
1.5 ≈ 2.0532
Error: |2.0532 − 2.0801 | = |−0.0269 | = 0.0269
=.=GQ
Percent Error:
= 0.013 = 1.3%
.=
7. , 0 = 0.5; 0.5
Solution:
x(n) y(n)
0.0 0.5000
0.1 0.5215
0.2 0.5362
0.3 0.5449
0.4 0.5490
0.5 0.5503
Solution to ODE: =
G
+−1
0.5 ≈ 0.5493
Error: |0.5493 − 0.5503 | = |−0.0010 | = 0.0010
=.==+=
Percent Error:
= 0.002 = 0.2%
=.Q
) .,