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MATH 215/255 Assignment 2
Submit online via Canvas by 6pm, Tuesday, September 28, 2021
§1.4, §1.6, §1.7, §1.8
1. (Exercise 1.4.11) Initially 5 grams of salt are dissolved in 20 liters of water. Brine with concen-
tration of salt 2 grams of salt per liter is added at a rate of 3 liters a minute. The tank is mixed
well and is drained at 3 liters a minute. How long does the process have to continue until there
are 20 grams of salt in the tank?
2. Consider the equation,
dy
dt
= cos(4y),
(a) Find all critical points.
(b) Classify their stability.
(c) What range of initial conditions results in solutions with limt→∞ y(t) = pi8 . What about
limt→∞ y(t) = −3pi8 and limt→∞ y(t) =
3pi
8 ?
3. Consider the equation,
dy
dt
= ay − y2,
(a) Again consider the cases a < 0, a = 0, and a > 0. In each case find the critical points,
draw the phase line, and determine whether each critical point is asymptotically stable,
metastable(semi-stable), or unstable.
(b) In each case sketch the direction field and several solutions of the equation in the ty-plane.
4. A second-order chemical reaction involves the interaction of one molecule of a substance P with
one molecule of a substance Q to produce one molecule of a new substance X; this is denoted
by P +Q→ X. Suppose that p and q, where p 6= q, are the initial concentrations of P and Q,
respectively, and let x(t) be the concentration of X at time t. Then p − x(t) and q − x(t) are
the concentrations of P and Q at time t, and the rate at which the reaction occurs is given by
the equation
dx
dt
= α(p− x)(q − x),
where α is a positive constant.
(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 equation is replaced by
dx
dt
= α(p− x)2.
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 determine x(t) for any t.
5. Find the general solution for
2x+ y2 + 2xyy′ = 0
6. Find the integrating factor for the following equation and then solve it,
(x+ 2)siny + (xcosy)y′ = 0, y(1) =
pi
2
.
7. Solve
1 + (
x
y
− siny)y′ = 0
M1. The logistic equation with growth rate r and carrying capacity K is given by
y′ = ry
(
1− y
K
)
The general solution is the logistic function
y(t) =
Ky0e
rt
K + y0(ert − 1)
where y0 = y(0). The following script plots the logistic function on the interval 0 ≤ t ≤ 5 for
certain values of r, K and y0:
t = 0:0.02:5;
r = 0.8; K = 2; y0 = 4;
y = K*y0*exp(r*t)./(K + y0*(exp(r*t) - 1));
plot(t,y)
Create a new script and copy/paste/modify the MATLAB code above into the new script to
plot in the same figure the logistic function on the interval 0 ≤ t ≤ 10 for the values:
• r = 1, K = 1, y0 = 2
• r = 0.5, K = 1, y0 = 2
• r = 1, K = 1, y0 = 1/2
• r = 0.5, K = 1, y0 = 1/2
Note that the command hold on allows you to plot multiple lines in a single figure. Save and
submit the figure as hw2M1.fig.
M2. The script hw2M2.m plots the approximation by Euler’s method of the equation y′ = y, y(0) = 1,
over the interval 0 ≤ t ≤ 2 with step size h = 0.2. The script simultaneously plots the exact
solution and computes the error of the approximation at tf = 2. Modify the script to plot:
• the approximation by Euler’s method of the equation y′ = −ty, y(0) = 1, over the interval
0 ≤ t ≤ 1 with step size h = 10−k such that k is the smallest (positive) integer such that
the error at tf = 1 is less than 0.005
• the exact solution of the equation y′ = −ty, y(0) = 1, over the interval 0 ≤ t ≤ 1
Run the script. Save and submit the figure as hw2M2.fig