Differential Equations/Linear Algebra Study Guide for Final
Typically, the final exams have 8 to 10 questions for you to solve. Since you have
not been tested on hw #7, my guess is that 20-30% of the final will be on the new
material. Outline below serves a guide to the concepts. Go over areas you feel weakest
in and practice, practice, and practice with the Final exams available to you! I wouldn’t
concentrate on to many hw problems (with the exception of hw#7).
Problems
1. Two to three problems will involve solve first order and second order DE and
initial value problems. Be able to classify the differential as indicated in review on
p.1. (Section 7.1 p.375 hw problems 1, 2, 3) Good also to know ExistenceUniqueness Theorem and how to apply for IVP (Section 1.5 and p.205)
A. First Order DE
i.
Separation of variables technique (Section 1.3)
ii.
Integrating Factor Method (Section 2.2)
iii.
Variation of Parameters (Section 2.2)
B. Second Order DE
i.
Finding Homogeneous Solution using the Characteristic
Equation (Section 4.1-4.3 - Summary Table p. 224)
ii.
Finding Particular Solution with Variation of Parameters,
(Section 4.4, Section 8.2-My guess is that there will be a
problem similar to those of section 8.2)
iii.
Reduction of order (See handout to review Cases I-IV)
2. One problem will most likely ask basic linear algebra questions. Given a matrix
A, you should be able to do the following:
A. Compute the determinant (Section 3.4)
B. Determine uniqueness, consistency of matrix (Section 3.2)
C. Put the matrix in RREF (Section 3.2)
D. Finding eigenvalues, eigenvectors (See handouts and Section 5.3)
E. Determine whether a set of vectors are linearly independent, span a vector
space (Section 3.5)
3. Solving System of DE
A. Be able to rewrite a higher order DE into a system of first order DE
(Section 6.1)
B. Be able to find the general solution of the system of DE (Sections 6.2-6.3)
C. Be able to find the generalized eigenvector in a case a double eigenvalue
occurs and you have a deficient eigenspace. (Section 6.2)
4. One problem may be a word type problem involving particular DE.
A. Growth model (Section 2.3)
B. Mixing/Cooling Models (Section 2.4)
C. Simple Harmonic Oscillator (Section 4.1)
5. Typically, there is a true/false problem on final testing your understanding of the
basic definitions, rules, etc. The best way to prepare for this section is to look at
problems of Final Exams and going over True/False sections from this semester’s
exams. Another possibility is a matching section involving direction fields.
6. One problem may ask you to classify the stability of the equilibrium solutions
(Section 6.5). Be able to draw phase portraits or phase planes. Review hw
problems on p.365-366 #1, 3, 4, 10
7. Another problem will mostly be on Linearization problem… (Section 7.2) .. You
most likely will be asked find the equilibrium solutions of nonlinear equations and
classify the equilibrium points (Table on p.384 is good to review). Review hw
problems p. 386-387 #1, 3, 4, 8, 9
8. Euler’s Method was asked on one final. You may want to review Section 1.4 and
look at hw problems p. 33 #1, 3