Math 456 Spring 2017. Problem Set 2.
(1) Consider the matrix
✓
A=
1
0
◆
and let P be the matrix defined by P = A
I. Then,
(a) Show by direct computation that P 2 = 0.
(b) Use the above fact to write etP explicitly.
(c) Use the above to write etA explicitly.
Hint: For c) you may use the following property of the exponential: if two matrices are such
that AB = BA, then eA+B is equal to the product eA eB .
(2) Let f (x1 , x2 ) = x41 + x42 + x62 . Show that
f = o(|x1 |2 + |x2 |3 ).
(3) Let f (x1 , x2 ) = x21 + x2 and let ↵ 2 (0, 1). Show that
f = o((x21 + x2 )↵ ).
(4) Let f (x1 , x2 ) = |x1 x2 |. Show that
f =o
✓q
◆
x21 + x22 .
Hint: Use the arithmetic-geometric mean inequality which states that 2|ab|  a2 + b2 , and
combine this with the previous problem.
(5) Compute the Taylor polynomial of degree five for f (x) = sin(x) at x = 0 and the Taylor polynomial of degree six for f (x) = cos(x) at x = 0. Then use a numerical software to plot the
di↵erence between each respective function and polynomial (for this last part, note that the
google search bar has some basic graphic calculators, you may also use matlab, the wolfram
alpha website, or any other software you like).
(6) Consider the Lotka-Volterra system
v(x1 , x2 ) = (↵x1
x1 x 2 ,
x1 + x 1 x 2 )
In terms of the parameters ↵, , , compute the two equilibrium points of this system. Compute also the derivative of v at each of the two fixed points and find their eigenvalues. Then,
discuss: is it reasonable to expect periodic solutions to the original system?.
(7) Given any di↵erentiable function U : R 7! R, consider the two dimensional system ẋ = v(x)
given by
ẋ1 = x2
ẋ2 =
U 0 (x1 )
Show that if (x1 (t), x2 (t)) is a solution to this system, then its trajectory lies along a curve in
the (x1 , x2 ) given by the equation
1 2
x + U (x1 ) = C for some C 2 R.
2 2
Bonus: Discuss how the above may be used to decide based on U whether the above system
admits periodic solutions.
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