Download Mathematics 351 - UD Math

Department of Mathematical
Sciences
University of Delaware
Prof. T. Angell
February 21, 2007
Mathematics 351
Exercise Sheet 2
Exercise 6: Show that if a and λ are positive constants, and b is any real number, then
every solution of the equation ẋ + ax = b exp (−λ t) has the property that x(t) → 0 as
t → ∞. (HINT: Consider the cases a = λ and a 6= λ separately.)
Exercise 7: Show that the functions y1 (x) ≡ 0, and y2 (x) = (t − to )3 are both solutions
of the initial value problem y 0 = 3 y 2/3 , y(to ) = 0 on −∞ < x < ∞. Graph both solutions
for the successive values of to = −1, 0, 1.
Exercise 8: A skydiver weighing 180 lbs. (including equipment) falls vertically downward
froman altitude of 5000 ft., and opens the parachute after 10 sec. of free fall. Assume
that the force of air resistance is 0.75|v| when the parachute is closed and 12|v| when the
parachute is open, where v is the velocity is measured in feet per second.
(a) Find the speed of the skydiver when the parachute opens.
(b) Find the distance fallen before the parachute opens.
(c) Show that limt→∞ v(t) := vL is finite. What is this “limiting velocity” vL after the
parachute opens?
(d) Determine how long the skydiver is in the air after the parachute opens.
Exercise 9: The equation ẋ = α(p − x)(q − x), α > 0 is a model for a second order
chemical reaction of species P and Q (x(t) is the concentration of the reaction product
X at time t, p is the initial molar concentration of P and q for Q).
(a) If x(0) = 0, determine the limiting value of x(t) as t → ∞ WITHOUT solving the
differential equation. Then solve the initial value problem and find x(t) for any t.
(b) If the substances P and Q are the same, then p = q and the kinetic equation is
replaced by ẋ = α(p − x)2 . If x(0) = 0, determine the limiting value of x(t) as
t → ∞ without solving the differential equation.
Exercise 10: An equation of the form y 0 + p(t)y = q(t)y n , is called a Bernoulli equation.
(a) Solve Bernoulli’s equation when n = 0 and when n = 1.
(b) Show that if n 6= 0, 1 then the substitution v = y 1−n reduces Bernoulli’s equation
to a linear equation.
(c) Solve t2 y 0 + 2ty − y 3 = 0, t > 0.
Due Date: Wednesday, Feb. 28, 9:05 AM