MATH 215/255
Fall 2014
Assignment 2
§1.4, §Exact equations ([Braun’s Section 1.9]), §1.6
Solutions to selected exercises can be found in [Lebl], starting from page 303.
• 1.4.8: Solve
x2
1
y 0 + xy = 3 with y(0) = 0.
+1
• 1.4.10: Newton’s law of cooling states that dx
dt = −k(x − A) where x is the temperature, t is time, A is the ambient temperature, and k > 0 is a constant. Suppose
that A = A0 cos(ωt) for some constants A0 and ω. That is, the ambient temperature
oscillates (for example night and day temperatures).
a) Find the general solution.
b) In the long term, will the initial conditions make much of a difference? Why or
why not?
• 1.4.102: Solve y 0 + 2 sin(2x)y = 2 sin(2x), for y(π/2) = 3.
• 1.4.103: Suppose a water tank is being pumped out at 3 L/min. The water tank
starts at 10 L of clean water. Water with toxic substance is flowing into the tank at
2 L/min, with concentration 20 t g/L at time t. When the tank is half empty, how
many grams of toxic substance are in the tank (assuming perfect mixing)?
• Ex 6 (Braun): Find the general solution of the differential equation
y2
dy
− 2yet + (y − et )
= 0.
2
dt
• Ex 8 (Braun): Solve the initial-value problem:
2t cos y + 3t2 y + (t3 − t2 sin y − y)
dy
= 0, y(0) = 2.
dt
• Ex 14 (Braun): Determine the constant a so that the following equation is exact,
and then solve the resulting equation:
eat+y + 3t2 y 2 + (2yt3 + eat+y )
dy
= 0.
dt
• Ex 18 (Braun): The differential equation f (t)(dy/dt) + t2 + y = 0 is known to have
an integrating factor µ(t) = t. Find all possible functions f (t).
• 1.6.4: Take x0 = sin x. a) Draw the phase diagram for −4π ≤ x ≤ 4π. On this
interval mark the critical points stable or unstable. b) Sketch typical solutions of the
equation. c) Find limt→∞ x(t) for the solution with the initial condition x(0) = 1.
• 1.6.5: Suppose f (x) is positive for 0 < x < 1, it is zero when x = 0 and x = 1, and it
is negative for all other x. a) Draw the phase diagram for x0 = f (x), find the critical
points, and mark them stable or unstable. b) Sketch typical solutions of the equation.
c) Find limt→∞ x(t) for the solution with the initial condition x(0) = 0.5.
2