Download Ordinary Differential Equations I – Review questions First order

Centre for Mathematical Sciences
Mathematics, Faculty of Science
Ordinary Differential Equations I – Review questions
At the oral exam you will get questions from the list below, but I might reformulate them and ask
follow-up questions. I’ve tried to indicate where you can find the relevant theory. AA stands for
Ahmad & Ambrosetti, W1 for the handout on the matrix exponential and W2 for the handout on
numerical methods. In a couple of cases you have to look in the lecture notes. The questions marked
with a ‘?’ are more difficult and are mainly intended for those of you who want a ‘VG’ (pass with a
distinction). Note that you are still expected to know what the relevant theorems say, but the proofs
are for those aiming for a ‘VG’.
First order equations
AA 1, 3. These are not questions for the oral exam, but you should review them for the written exam.
• How do you solve a first order linear differential equation?
• How do you solve a separable equation?
• What is an exact equation? How do you check if an equation is exact?
• What is an integrating factor for an equation of the form M (x, y) dx + N (x, y) dy = 0?
Existence and uniqueness of solutions
AA 2, 4.
1. Give an example of an initial value problem with more than one solution.
2. What does it mean for a function f (t, x) to be (locally) Lipschitz at a point (t0 , x0 ) with respect
to x? What does it mean for f to be globally Lipschitz?
3. ? Formulate and prove a theorem about existence and uniqueness for a first order equation (Theorem 2.4.5 suffices for the proof, but you should also know what Theorem 2.4.4 says and have
some idea of how it follows from Theorem 2.4.5).
Remark: everyone should know the statements of the theorems, but the proof is marked with a
star.
4. Discuss existence and uniqueness for systems and higher order equations.
Please, turn over!
Second and higher order linear equations
AA 5, 6.
5. Show that the solution space of an nth order homogeneous linear equation is n-dimensional.
(Lecture notes, Sep 14 and 17)
6. ? Show that one can find a fundamental set of solutions of the form tk emi t for an nth order
homogeneous linear equation with constant coefficients. (Lecture notes, Sep 17)
7. What is the Wronskian? Prove Abel’s theorem for a second order equation.
8. Derive the variation of parameters formula for a nonhomogeneous second order linear equation.
Linear systems
AA 7, W1.
9. Define etA when A is a square matrix. Motivate why the series converges.
10. State and prove the Cayley-Hamilton theorem.
11. What is meant by a generalized eigenvector?
12. ? Show that Cn has a basis consisting of generalized eigenvectors.
13. How do you compute etA using a basis of generalized eigenvectors?
14. Show that the set of solutions of a linear homogeneous first order system with n unknowns is a
vector space of dimension n. (Lecture notes, Sep 21)
15. Derive the variation of parameters formula for a nonhomogeneous first order linear system.
The Laplace transform
AA 11.
16. Define the Laplace transform. Show that it is well-defined for piecewise continuous functions
of exponential order.
17. Show that L{f 0 }(s) = sL{f }(s) − f (0). How is this used when solving differential equations
with constant coefficients?
Numerical methods
W2.
18. Describe Euler’s method for the numerical solution of differential equations. Derive local and
global error estimates.
Power series solutions
AA 10.
19. What is an ordinary point of the equation a0 (t)x(n) +a1 (t)x(n−1) +· · ·+an−1 (t)x0 +an (t)x =
0? What is a regular singular point?
20. ? State and prove a theorem about power series solutions at an ordinary point (for the proof it
suffices to take n = 2). (Lecture notes, Oct 15 and 19)
Remark: everyone should know the statement of the theorem, but the proof is marked with a
star.
21. Describe Frobenius’ method for finding solutions near a regular singular point. In particular,
define the indicial equation and discuss when there are two linearly independent Frobenius series
solutions.