Anthony Kitsmiller Sharma MAT 275 ONLINE B Fall 2015
Assignment Test 1 due 11/05/2015 at 11:45am MST
Problem 1. 1. (2 pts) Find the value of kfor which the con-
stant function x(t)=kis a solution of the differential equation
4t3dx
dt +4x+2=0.
Answer(s) submitted:
•-1/2
(correct)
Correct Answers:
•-0.5
Problem 2. 2. (3 pts) Find the particular solution of the dif-
ferential equation
dy
dx +ycos(x)=4cos(x)
satisfying the initial condition y(0)=6.
Answer: y=
Your answer should be a function of x.
Answer(s) submitted:
•2eˆ(-sin(x))+4
(correct)
Correct Answers:
•4 + 2*eˆ(-sin(x))
Problem 3. 3. (3 pts) Find the function satisfying the differ-
ential equation
y−4y=−9e1t
and y(0)=−4.
Answer(s) submitted:
•3eˆt-7eˆ(4t)
(correct)
Correct Answers:
•3*exp(1*t) + -7*exp(4*t)
Problem 4. 4. (3 pts)
Find the solution to the differential equation dy
dx =19xy
(lny)10
which passes through the point (0,e). Express your answer as
lny=
Answer(s) submitted:
•(((209xˆ2)/2)+1)ˆ(1/11)
(correct)
Correct Answers:
•(19*(10+1)*x*x/2 + 1)ˆ(1/(10+1))
Problem 5. 5. (3 pts) Solve the following initial value prob-
lem:
tdy
dt +9y=6t
with y(1)=5.
Put the problem in standard form.
Then find the integrating factor, ρ(t)= ,
and finally find y(t)= .
Answer(s) submitted:
•tˆ9
•((3t)/5)+(22/5)tˆ(-9)
(correct)
Correct Answers:
•tˆ(9)
•(6*t/(1+9)) + 4.4 * (t**(-9 ))
Problem 6. 6. (6 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.
?
?
?
?
Answer(s) submitted:
•-3
•stable
•-1
1
•semi-stable
•.5
•unstable
•1.5
•stable
(correct)
Correct Answers:
•-3
•STABLE
•-1
•SEMI-STABLE
•0.5
•UNSTABLE
•1.5
•STABLE
Problem 7. 7. (5 pts) A population Pobeys the logistic
model. It satisfies the equation
dP
dt =7
1300 P(13 −P)for P>0.
(a) The population is increasing when <P<
(b) The population is decreasing when P>
(c) Assume that P(0)=2.Find P(74).
P(74)=
Answer(s) submitted:
•0
•13
•13
•(7/1300)-74(13-74)
(score 0.75)
Correct Answers:
•0
•13
•13
•12.6096795348195
Problem 8. 8. (4 pts) Use Euler’s method with step size
0.5 to compute the approximate y-values y1≈y(1.5),y2≈y(2),
y3≈y(2.5),and y4≈y(3)of the solution of the initial-value
problem
y=2+4x−2y,y(1)=4.
y1=,
y2=,
y3=,
y4=.
Answer(s) submitted:
•
•
•
•
(incorrect)
Correct Answers:
•3
•4
•5
•6
Problem 9. 9. (3 pts) A tank contains 2520 L of pure water.
Solution that contains 0.06 kg of sugar per liter enters the tank
at the rate 8 L/min, and is thoroughly mixed into it. The new
solution drains out of the tank at the same rate.
(a) How much sugar is in the tank at the begining?
y(0)= (kg)
(b) Find the amount of sugar after t minutes.
y(t)= (kg)
(c) As t becomes large, what value is y(t)approaching ? In
other words, calculate the following limit. lim
t→∞
y(t)= (kg)
Answer(s) submitted:
•0
•151.2-151.2ˆ((1t)/315)
•151.2
(score 0.666670024394989)
Correct Answers:
•0
•151.2*(1-2.71828182845905ˆ(-0.00317460317460317*t))
•151.2
Problem 10. 10. (8 pts)
Determine which differential equation corresponds to
each phase line. You should be able to state briefly how
you know your choices are correct.
? 1. dy
dt =3y−y2
? 2. dy
dt =y2−3y
? 3. dy
dt =y3−4y
? 4. dy
dt =y(y−2)
? 5. dy
dt =y(2−y)2
? 6. dy
dt =y2|y−2|
? 7. dy
dt =4y−y3
? 8. dy
dt =2y−y2
2
ABCD
EFGH
Answer(s) submitted:
•A
•C
•H
•B
•D
•F
•E
•G
(incorrect)
Correct Answers:
•A
•C
•B
•F
•D
•E
•H
•G
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