Math 267:
Build-Up Review for the Final
Finale Exam Review
Supplemental Instruction
Iowa State University
Leader: Ron
Course: Math 267
Instructor: Castillo-Gil, Hentzel, Sacks
Date: December 7th, 2015
1. Identify the standard differential equations for the following mathematical model types
and identify the sign of the constant:
a) What is the differential equation for population growth? What is the sign of the constant?
b) What is the differential equation for Newton’s Law of Cooling? What is the sign of the
constant?
c) What is the differential equation for radioactive decay? What is the sign of the constant?
d) What is the differential equation for a falling body without air resistance when the
acceleration of gravity is g?
2. Solve the differential equation: (𝑥 + 1) 𝑦 ′ + 𝑥𝑦 = 𝑒 −𝑥
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3. Solve the following equation and corresponding IVP. (Hint: Test for exactness first.)
(𝑦𝑐𝑜𝑠(𝑥𝑦) + 𝑒 𝑥−𝑦 )𝑑𝑥 + (𝑥𝑐𝑜𝑠(𝑥𝑦) − 𝑒 𝑥−𝑦 + 1)𝑑𝑦 = 0; 𝑦(√𝜋) = √𝜋
4. Solve the following equation using an appropriate substitution:
𝑑𝑦
− 2𝑥𝑦 = 𝑥𝑦 3
𝑑𝑥
5. Use undetermined coefficients to find yp for the following differential equation:
𝑦 ′′ + 𝑦 ′ + 𝑦 = 2𝑥𝑒 𝑥 + 3𝑒 𝑥
6. Use variation of parameters to find yp for the following differential equation:
𝑦 ′′ − 2𝑦 ′ + 2𝑦 = 𝑒 𝑥 csc(𝑥)
7. A mass weighing 8 pounds is attached to a spring. When set in motion, the spring/mass
system exhibits simple harmonic motion. Determine the equation of motion if the
spring constant is 1 lb/ft and the mass is initially released from a point 6 inches below
3
the equilibrium position with a downward velocity of 2 ft/s.
8. Consider the differential equation: (𝑥 2 − 16)𝑦 ′′ + 𝑦 = 0; 𝑦(0) = 64, 𝑦 ′ (0) = 0
a) Find the minimum radius of convergence R, of a power series solution of the given
differential equation about the ordinary point 𝑥 = 0.
𝑛
b) Let the solution be ∑∞
𝑛=0 𝑐𝑛 𝑥 . Find a recurrence relation for the coefficients of the power
series, and explicitly give the first 4 terms of the series, that is, provide the truncated
series as a polynomial of order 4.
9. Find the given inverse Laplace transform:
𝐿−1 {
2𝑠 − 6
}
𝑠2 + 9
10. Find the Laplace transform of the given differential equation below with the given
initial-value condition:
𝑦 ′ + 6𝑦 = 𝑒 4𝑡 , 𝑦(0) = 2
11. Find either f(t) or F(s) as indicated below:
a. 𝐿{𝑡 3 𝑒 −2𝑡 }
b. 𝐿{𝑡𝑈(𝑡 − 2)}
12. Find the general solution of the following system using eigenvalues/eigenvectors
method:
3
2
𝑥⃗ ′ = (
−18
) 𝑥⃗
−9
13. Find the general solution of the following system using eigenvalues/eigenvectors
method:
−2 −4 2
[−2 1 2]
4
2 5
14. Use the method of undetermined coefficients to find a general solution to the system
below:
𝑥⃗ ′ (𝑡) = [
−𝑡 − 1
1 1
] 𝑥⃗(𝑡) + [
]
−4𝑡 − 2
4 1
15. Use the method of variation of parameters to find the general solution to the system:
4
0 3
𝑥⃗ ′ = (
) 𝑥⃗ + ( ) 𝑒 𝑡
−1 4
−4
16. Write the given nonlinear second-order differential equation as a plane autonomous
system. Find all critical points of the resulting system.
𝑥 ′′ + 𝑥 ′ (1 − 𝑥 3 ) − 𝑥 2 = 0