Mathematics 118A, Winter 1998

Homework Assignment 2, due Friday, January 23

1. Let p(z) be a polynomial with real coefficients. Prove that if z ∈ ℂ is a zero of p, then its complex conjugate \overline{z} is also a zero of p.

2. (a) Prove that for an n × n matrix,

\frac{d}{dt}e^{At} = Ae^{At}.

(b) Prove explicitly that the solution x = x(t) to

ẋ = Ax,  x(0) = x_{0},  (*)

is unique. Hint: Let x and y denote two solutions to (*). Derive a differential equation for the (scalar) quantity w(t) = ∥x(t) − y(t)∥^{2}.

(c) Prove that any solution to (*) can be written in the form x(t) = e^{At}x_{0}.

(d) Show that if two n × n matrices A and B commute, i.e., if AB = BA, then

Be^{At} = e^{At}B.

Hint: Verify that Be^{At}x_{0} and e^{At}Bx_{0} solve ẋ = Ax, then use uniqueness.

(e) Show that if A and B commute, then

e^{(A+B)t} = e^{At}e^{Bt}.

Hint: Use the result from part (d) to show that both sides are one parameter groups of transformations that solve the same differential equation (which one?).

(f) Use (e) to solve the differential equation

ẋ = Mx where M = (\begin{matrix}2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{matrix}) .

3. Consider the Lienard form of the Van der Pol equation,

ẋ_{1} = x_{2} − λ(\frac{1}{3}x_{1}^{3} − x_{1})
ẋ_{2} = −x_{1},

where λ is a parameter. Show that the only equilibrium point is at the origin. Determine its stability type as a function of λ.

4. Find all equilibrium points, determine their stability properties, and try to sketch the phase portraits of the following differential equations:

(a) \begin{aligned}ẋ_{1} & = 2x_{1} − x_{1}^{2} − x_{1}x_{2} \\ ẋ_{2} & = −x_{2} + x_{1}x_{2} \end{aligned}

(b) ÿ + ẏ + y^{3} = 0