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)1, 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.
1Unstable points with one of the arrows pointing towards the critical point are called metastable/semi-stable.
1
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
2
MATH 215/255 Assignment 1M
Submit online via Canvas by 6pm, Tuesday, September 28, 2019
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
3

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