Math 2250 Lab 1
Name/Unid:
1. (25 points) Consider the second order differential equation
d2 y 1 dy
= 0.
+ 1+
dx2
x dx
for x > 0.
(a) Verify that
Z
y(x) =
1
x
e−u
du
u
is a solution to the above ODE.
(b) Derive the solution yourself using the conditions y 0 (1) = 1/e and y(1) = 0 following
the given steps:
i. Multiply both sides of the differential equation by g(x) = ex+ln x . What do you
notice about the derivative g 0 (x)?
ii. Use (i) and the fact that
f 0 (x)eg(x) + f (x)g 0 (x)eg(x) =
d
f (x)eg(x)
dx
to convert the second order differential equation to a first order differential
equation by integration. Use y 0 (1) = 1/e to find the constant of integration.
iii. Once you have the first order differential equation, integrate from 1 to x and
use y(1) = 0 to find the solution.
2. (25 points) Consider the motion of a charged particle in an electric field. In the absence
of magnetic field and gravity, the force exerted on a particle with charge q due to an
electric field with strength E is given by
~
F~ = q E.
Suppose there are two plates of length ` that create a constant electric field pointing in
the negative y direction, that is
~ = −E0 ŷ.
E
Furthermore suppose that when the particle enters the plate region it is moving in the
positive x direction at a constant velocity vx,0 .
(a) Derive the equation of motion for the particle assuming that the particle starts at
x = 0, y = 0. That is, find x(t), y(t).
(b) Suppose that when the particle leaves the region it has been vertically displaced by
amount ∆y. If the mass of the particle is m, then find the charge. Assume that the
plates are far enough apart so that the particle actually leaves the region.
(c) Now assume that the particle is affected by gravity as well as the electric field
between the plates. Find x(t), y(t), and the charge of the particle, using the same
initial conditions and coordinates as in part (a).
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3. (25 points) Consider the time-dependent velocity of the car shown in the figure below.
(a) Calculate the acceleration a(t) for 0 ≤ t ≤ 10 except for at times t = 1, 5, 8. Why
are these points problematic?
(b) Use the Fundamental Theorem of Calculus to find the equation for the position x(t)
for 0 ≤ t ≤ 10 and graph x = x(t). Use x(0) = 0 and continuity of x(t) to find the
constants of integration.
(c) Use your equation for x(t) to determine how far the car has traveled at t = 6
minutes.
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4. (25 points) A model for the population growth of a species is given by
dy
= y(1 − y) − h
dt
where h > 0 is the natural death rate of the population.
Sketch a series of slope fields for h = 0, .2, .4. For each slope field draw an approximate
solution with initial conditions y(0) = 0, .2, .6, 1.2. Explain the differences in the solutions for the different values of h. For which values of h does the population persist or
go extinct? (By persist we mean that the population size is positive for all time and by
extinct we mean that the population size becomes negative and stays negative for all
future time.) Does this depend on initial population size?
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