Mathematics 304, section 2 – Ordinary Differential Equations Practice Final Exam December 13, 2004

I. Consider the ODE

y′ = \frac{y}{t^{2}}

A) SHow that the constant function y_{1}(t) = 0 is a solution. B) Show that there are infinitely many other functions that satisfy this equation, that

agree with y_{1}(t) for t ≤ 0, but that are nonzero when t > 0. C) Why doesn’t this example contradict the Existence and Uniqueness Theorem?

II. A student starting college places $ 500 into a bank account that pays interest at the rate of 4% per year, compounded continuously. The student also saves $ 520 over the course of each year from her job and has it deposited into the account continuously. Writing B(t) for the balance in the account as a function of t in years, these conditions say

B′ = (.04)B + 520

What will the balance in the account be at the end of 4 years?

III. Construct the bifurcation diagram for the family of ODE

y′ = y^{3} + ay^{2}

where a is an arbitrary real parameter. For which values of a do bifurcations occur?

IV. A) Show that if V is an eigenvector of A with eigenvalue λ, then X(t) = e^{λt}V is a solution

of X′ = AX. B) Find the solution of X′ = ( \begin{matrix}4 & 2 \\ 1 & 3 \end{matrix} ) X satisfying X(0) = ( \begin{matrix}1 \\ 0 \end{matrix} ).

C) Sketch the phase portrait for the system from B.

D) Find the solution of X′ = ( \begin{matrix}0 & 2 \\ −2 & −1 \end{matrix} ) X with X(0) = ( \begin{matrix}2 \\ 1 \end{matrix} ).

V. For the family of 1st order systems

X′ = ( \begin{matrix}a & 1 \\ a & a \end{matrix} ) ,

sketch the corresponding curve in the trace-determinant plane, and discuss any bifurcations that occur.

VI. A 3 × 3 matrix A has characteristic polynomial (1 − λ)^{3} What are the possible canonical forms C of A, how would you tell which canonical form A has, and what are the corresponding geneneral solutions of the systems Y′ = CY for each possible C?

VII. Consider the forced oscillator equation

y′′ + 5y′ + 6y = e^{−t} + 4

A) Compute the solution with y(0) = y′(0) = 0. B) What would change in your solution if the e^{−t} term on the right was replaced by e^{−2t}.

VIII. Consider the nonlinear first order system

\{ \begin{matrix}x′ = −2x + y \\ y′ = −y + x^{2} \end{matrix}

A) Find all the equilibrium points. B) Linearize at each equilibrium and classify the equilibrium of the linearized systems as

sources, sinks, saddles, centers, or degenerate. C) What can you say about the types of the equilibrium points of the non-linear system

based on B?

IX. Consider

\{ \begin{matrix}x′ = 1 − y^{2} \\ y′ = x(2 − y) \end{matrix}

A) Is this system Hamiltonian? Why or why not? B) Show that there is a function μ(x, y) such that the system

\{ \begin{matrix}x′ = μ(x, y)(1 − y^{2}) \\ y′ = μ(x, y)x(2 − y) \end{matrix}

is Hamiltonian. C) Find a conserved quantity for this new system, then show it is also conserved for the

original system.