Math 251
teddy123
Extra Credit Assignment for MATH 251, Spring 2017
Date due: Friday April 28 (in class)
Student:
1. (5 points) Consider the autonomous differential equation
y′ = 36y2 − 9y4.
(a) Find all of its equilibrium solutions:
(b) Classify the stability of each equilibrium solution (justify your answer):
(c) Suppose y1(t) and y2(t) are two solutions of the equation such that y1(17) = −1 and y2(29) = 1. What is lim
t→∞ (y1(t) + y2(t))?
(d) Suppose y(−1729) = β, and that lim t→∞
y(t) = 2. Find all possible values of β.
(e) If y(−53) = −2, then what is y(42)? Without solving the equation, briefly explain your conclusion.
2. (3 point) Consider the initial value problem:
(t2 − 1)y′ + ln(t)y = tan(2πt), y(0.875) = 10.
According to the existence and uniqueness theorem what is the largest interval on which a unique solution is guaranteed to exist?
3. (14 points) Find the inverse Laplace transform of the following function
e−4s 3s2 + 17s + 10
(s + 5)(s2 + 4s + 5)
4. (10 points) Find the general solution to the following linear system
x′ =
[ 2 1 1 1
] x
5. (16 points) Let f(x) =
{ 1 for 0 < x < 1,
2 − x for 1 < x < 2,
(a) Consider the odd periodic extension, of period T = 4, of f(x). Sketch 3 periods, on the interval −6 < x < 6, of this function.
(b) To which value will the Fourier series of the odd periodic extension converge to at x = 2?
(c) Now consider the even periodic extension, of period T = 4, of f(x). Sketch 3 periods, on the interval −6 < x < 6, of this function.
(d) Find the constant term a0 2
of the Fourier series of this even periodic extension.