MAP 2302 Exam #1
Name:
ID#
HONOR CODE: On my honor, I have neither given nor received any aid on this
examination.
Signature:
Instructions: Do all scratch work on the test itself. Make sure your final answers
are clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALL
WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT!!!
No.
1
2
3
4
5
6
Total
Score
/14
/18
/18
/18
/18
/14
/100
(1) Verify by substitution that the given function is a solution of the given differential
equation. Primes denote derivatives with respect to x. (14 points)
y 00 − 6y 0 + 13y = 0; y = e3x cos 2x
(2) Find a function y = f (x) satisfying the given differential equation and the prescribed initial condition. (18 points)
dy
x+5
= 2
; y(2) = 4 − ln 4
dx
x +x−2
(3) Determine whether or not the Existence and Uniqueness Theorem guarantees
the existence of a solution of the given initial value problem. If existence is
guaranteed, determine whether Existence and Uniqueness Theorem does or does
not guarantee uniqueness of that solution. (6 points each)
2
(a)
dy
dx
= xy 5 ; y(1) = 0
(b)
dy
dx
= x2 + ln y; y(−4) = −1
(c)
dy
dx
= cos y; y(0) = π
(4) Find a general explicit solution of the differential equation. Be sure to simplify
as much as possible. (18 points)
dy
= e3x+2y
dx
(5) Find the general solution to the differential equation. Primes denote derivatives
with respect to x. (18 points)
xy 0 − 4y = x6 ex
(6) Let P (t) represent the size of a population at time t. The logistic equation is a
differential equation which can be used to model the way the population grows.
The equation is
dP
P
= rP 1 −
; P (0) = P0
dt
K
where r represents the populations natural growth rate, K represents the environmental carrying capacity (the number of individuals that the environment can
sustain), and P0 is the initial population size. To solve this equation, we can
P
simplify it by making the substitution x = K
. This leads to the equation
dx
= rx(1 − x).
dt
(a) Solve this equation for x(t). (Note: Remember that r is just an arbitrary
constant, which depends on the specific population we are modeling.) (10
points)
(b) Find the solution, P (t) to the logistic equation by using the substitution
P
, the initial condition P (0) = P0 , and your answer in part (a). (4
x = K
points)
(c) Bonus: If P > K (i.e., if the population size is greater than the environmental
carrying capacity), what does the logistic equation tell you will happen to
the population’s size? If P < K (i.e., if the population size is less than the
environmental carrying capacity), what does the logistic equation tell you will
happen to the population’s size? Explain your answers. (6 points)
(d) Bonus: Based on the solution to the logistic equation, determine the eventual
fate of the population. In other words, determine lim P (t). (4 points)
t→∞
(6) continued.
Scratch Paper
Scratch Paper