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
3. Solve the following equation and corresponding IVP. (Hint: Test for exactness first.)
ycosxy+ ex-ydx+xcosxy-ex-y+1dy=0;yπ= π
4. Solve the following equation using an appropriate substitution:
dydx-2xy=xy3
5. Use undetermined coefficients to find yp for the following differential equation:
y''+y'+y=2xex+3ex
6. Use variation of parameters to find yp for the following differential equation:
y''-2y'+2y=excsc(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 the equilibrium position with a downward velocity of 32 ft/s.
8. Consider the differential equation: x2-16y''+y=0;y0=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.
b) Let the solution be n=0∞cnxn. 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-12s-6s2+9
10. Find the Laplace transform of the given differential equation below with the given initial-value condition:
y'+6y=e4t, y0=2
11. Find either f(t) or F(s) as indicated below:
a. Lt3e-2t
b. LtU(t-2)
12. Find the general solution of the following system using eigenvalues/eigenvectors method:
x'=3-182-9x
13. Find the general solution of the following system using eigenvalues/eigenvectors method:
-2-42-212425
14. Use the method of undetermined coefficients to find a general solution to the system below:
x't=1141xt+-t-1-4t-2
15. Use the method of variation of parameters to find the general solution to the system:
x'=03-14x+4-4et
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-x3-x2=0