Anthony Kitsmiller Sharma MAT 275 ONLINE B Fall 2015
Assignment Midterm due 11/11/2015 at 01:35pm MST
Problem 1. 1. (5 pts) Find the particular solution of the dif-
ferential equation
dy
dx +ycos(x)=5cos(x)
satisfying the initial condition y(0)=7.
Answer: y=
Your answer should be a function of x.
Answer(s) submitted:
•5+2eˆ(-sin(x))
(correct)
Correct Answers:
•5 + 2*eˆ(-sin(x))
Problem 2. 2. (5 pts) Solve the following initial value prob-
lem:
tdy
dt +4y=3t
with y(1)=9.
(Find yas a function of t.)
y=
Answer(s) submitted:
•(3(tˆ5+14))/(5tˆ4)
(correct)
Correct Answers:
•(3*t/(1+4)) + 8.4 * (t**(-4 ))
Problem 3. 3. (5 pts) A tank contains 2340 L of pure water.
A solution that contains 0.08 kg of sugar per liter enters tank at
the rate 2 L/min The solution is mixed and drains from the tank
at the same rate.
(a) How much sugar is in the tank at the beginning.
y(0)= (include units)
(b) Find the amount of sugar (in kg) after tminutes.
y(t)= (function of t)
(b) Find the amout of the sugar after 72 minutes.
y(72)= (include units)
Answer(s) submitted:
•0kg
•187.2-187.2eˆ(-t/1170)
•187.2-187.2eˆ(-72/1170)kg
(correct)
Correct Answers:
•0kg
•187.2*(1-eˆ(-0.000854700854700855*t))
•11.1726989685872 kg
Problem 4. 4. (5 pts) Use Euler’s method with step size
0.5 to compute the approximate y-values y1≈y(0.5),y2≈y(1),
y3≈y(1.5),and y4≈y(2)of the solution of the initial-value
problem
y=2+3x+2y,y(0)=−4.
y1=,
y2=,
y3=,
y4=.
Answer(s) submitted:
•-6
•-8.75
•-19.5
•-35.5
(incorrect)
Correct Answers:
•-7
•-12.25
•-22
•-40.75
Problem 5. 5. (5 pts) The graph of the function f(x)is
(the hori-
zontal axis is x.)
Given the differential equation x(t)= f(x(t)).
List the constant (or equilibrium) solutions to this differential
equation in increasing order and indicate whether or not these
equations are stable, semi-stable, or unstable.
?
?
1
?
?
Answer(s) submitted:
•-3
•stable
•-1
•semi-stable
•1
•unstable
•3
•stable
(correct)
Correct Answers:
•-3
•STABLE
•-1
•SEMI-STABLE
•1
•UNSTABLE
•3
•STABLE
Problem 6. 6. (5 pts)
Newton’s law of cooling says that the rate of cooling of an
object is proportional to the difference between the temperature
of the object and that of its surroundings (provided the differ-
ence is not too large).
If T=T(t)represents the temperature of a (warm) object
at time t,Arepresents the ambient (cool) temperature, and kis
a negative constant of proportionality, which equation(s) accu-
rately characterize Newton’s law?
•A. dT
dt =k(T−A)
•B. dT
dt =kT(1−T/A)
•C. dT
dt =k(A−T)
•D. dT
dt =kT(T−A)
•E. All of the above
•F. None of the above
Answer(s) submitted:
•A
(correct)
Correct Answers:
•A
Problem 7. 7. (5 pts) Suppose that a population develops
according to the logistic equation
dP
dt =0.15P−0.0002P2
where tis measured in weeks.
(a) What is the carriying capacity?
(b) Is the solution increasing or decreasing when Pis between 0
and the carriying capacity? ?
(c) Is the solution increasing or decreasing when Pis greater
than the carriying capacity? ?
Answer(s) submitted:
•750
•increasing
•decreasing
(correct)
Correct Answers:
•750
•INCREASING
•DECREASING
Problem 8. 8. (5 pts) Consider the first order differential
equation
y+t
t2−4y=et
t−8.
For each of the initial conditions below, determine the largest
interval a<t<bon which the existence and uniqueness the-
orem for first order linear differential equations guarantees the
existence of a unique solution.
(1) y(−4)=6.4.
(2) y(−0.5)=−5.5.
(3) y(0)=0.
(4) y(3.5)=−0.5.
(5) y(9)=−5.5.
Answer(s) submitted:
•t<-2
•-2<t<2
•-2<t<2
•2<t<8
•t<8
(score 0.800000011920929)
Correct Answers:
•t<-2
•-2<t<2
•-2<t<2
•2<t<8
•8<t
2
Problem 9. 9. (5 pts) Find the two values of kfor which
y(x)=ekx is a solution of the given differential equation.
y −8y+12y=0
smaller value =
larger value =
Answer(s) submitted:
•2
•6
(correct)
Correct Answers:
•2
•6
Problem 10. 10. (5 pts)
Find the general solution to the homogeneous differential equa-
tion
d2y
dx2+9dy
dx +14y=0
The solution has the form
y=C1f1(x)+C2f2(x)
with
f1(x)= and f2(x)=
Left to your own devices, you will probably write down the
correct answers, but in case you want to quibble, enter your an-
swers so that f1,f2are normalized with their value at x=0 equal
to 1.
Answer(s) submitted:
•exp(-7x)
•exp(-2x)
(correct)
Correct Answers:
•exp(-7*x)
•exp(-2*x)
Problem 11. 11. (5 pts) Find yas a function of tif
10000y +81y=0,
y(0)=8,y(0)=5.
y=
Answer(s) submitted:
•(500/9)sin(9t/100)+8cos(9t/100)
(correct)
Correct Answers:
•(8) *(1)*(cos((9/100)*t)) + (500/9) *(1)*(sin((9/100)*t))
Problem 12. 12. (5 pts) Determine whether the following
pairs of functions are linearly independent or not.
? 1. f(t)=tand g(t)=|t|
? 2. The Wronskian of two functions is W(t)=tare the
functions linearly independent or dependent?
? 3. f(θ)=12cos3θand g(θ)=48cos3θ−36cosθ
Answer(s) submitted:
•Linearly independent
•Linearly independent
•Linearly dependent
(correct)
Correct Answers:
•LINEARLY INDEPENDENT
•LINEARLY INDEPENDENT
•LINEARLY DEPENDENT
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