MATH 215/255
Fall 2014
Assignment 3
§1.6, §1.7, §2.1, §2.2
Solutions to selected exercises can be found in [Lebl], starting from page 303.
• 1.6.6: Start with the logistic equation dx
dt = kx(M − x). Suppose that we modify our
harvesting. That is we will only harvest an amount proportional to current population.
In other words we harvest hx per unit of time for some h > 0 (Similar to earlier
example with h replaced with hx). a) Construct the differential equation. b) Show
that if kM > h, then the equation is still logistic. c) What happens when kM < h?
• 1.6.101: Let x0 = (x − 1)(x − 2)x2 . a) Sketch the phase diagram and find critical
points. b) Classify the critical points. c) If x(0) = 0.5, then find limt→∞ x(t).
• 1.7.5: Approximate the value of e by looking at the initial value problem y 0 = y with
y(0) = 1 and approximating y(1) using Euler’s method with a step size of 0.2.
• 1.7.6: Example of numerical instability: Take y 0 = −5y, y(0) = 1. We know that the
solution should decay to zero as x grows. Using Euler’s method, start with h = 1 and
compute y1 , y2 , y3 and y4 to try to approximate y(4). What happened? Now halve
the interval. Keep halving the interval and approximating y(4) until the numbers you
are getting start to stabilize (that is, until they start going towards zero). Note: You
might want to use a calculator.
• 2.1.6: Suppose that (b − a)2 − 4ac > 0. a) Find a formula for the general solution of
ax2 y 00 + bxy 0 + cy = 0. Hint: Try y = xr and find a formula for r. b) What happens
when (b − a)2 − 4ac = 0 or (b − a)2 − 4ac < 0?
• 2.1.7: Same equation as in Exercise 2.1.6. Suppose (b − a)2 − 4ac = 0. Find a
formula for the general solution of ax2 y 00 + bxy 0 + cy = 0. Hint: Try xr ln x for the
second solution.
• 2.1.8: (Reduction of order) Suppose that y1 (x) is a solution of y 00 +p(x)y 0 +q(x)y = 0.
Find a second linearly independent solution y2 (x). Hint: Take y2 (x) = y1 (x)v(x), plug
it into equation and use the fact that y1 (x) is a solution to find v(x). It might be
useful to introduce w(x) = v 0 (x), and don’t forget about integration constants.
• 2.1.10: Take y 00 − 2xy 0 + 4y = 0. a) Show that y = 1 − 2x2 is a solution. b) Use
reduction of order to find a second linearly independent
R x t2 solution. Express your answer
2
√
in terms of imaginary error function erfi(x) = π 0 e dt. c) Write down the general
solution.
• 2.2.7: Find the general solution of y 00 + 9y 0 − 10y = 0.
• 2.2.103: Find the solution to 2y 00 + y 0 + y = 0, y(0) = 1, y 0 (0) = −2.