Victor Camacho
Assignment Homework 2 due 09/06/2012 at 11:00pm MDT
4.
1. (1 pt) Library/Michigan/Chap11Sec5/Q11.pg
Consider a 32◦ F object placed in 69◦ F room.
(a) Write a differential equation for H, the temperature of the
object at time t, using k > 0 for the constant of proportionality,
and write your equation in terms of H, k, and t.
H0 =
(b) Give the general solution for your differential equation.
Simplify your solution in terms of an unspecified constant C,
which appears as the coefficient of an exponential term, and the
growth factor k.
H=
(Your answer may involve the constant of proportionality k.)
(c) The temperature of the object is 32◦ F initially, and 37◦ F
one hour later. Find the temperature of the object after 5 hours.
H(5) =
degrees F
A thermometer is taken from a room where the temperature is
23oC to the outdoors, where the temperature is −11oC. After
one minute the thermometer reads 8oC.
(a) What will the reading on the thermometer be after 2 more
minutes?
,
(b) When will the thermometer read −10oC?
minutes after it was taken to the outdoors.
5.
(1 pt) Library/maCalcDB/setDiffEQ5ModelingWith1stOrder-
/ur de 5 4.pg
Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its
temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton’s law of cooling. If
the coffee has a temperature of 205 degrees Fahrenheit when
freshly poured, and 1.5 minutes later has cooled to 185 degrees
in a room at 64 degrees, determine when the coffee reaches a
temperature of 140 degrees.
The coffee will reach a temperature of 140 degrees in
minutes.
(a) What is the half-life of the element?
(days)
6. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/osu de 4 15.pg
Find the particular solution of the differential equation
(b) How long will it take for a sample of 100mg to decay to
82 mg?
time needed:
(1 pt) Library/maCalcDB/setDiffEQ5ModelingWith1stOrder-
/ur de 5 14.pg
2. (1 pt) Library/ma123DB/set7/s9 4 10.pg
An unknown radioactive element decays into non-radioactive
substances. In 420 days the radioactivity of a sample decreases
by 34 percent.
half-life:
math1220fall2012-2
dy
+ y cos(x) = 3 cos(x)
dx
(days)
satisfying the initial condition y(0) = 5.
Answer: y(x)=
.
3. (1 pt) Library/ma123DB/set7/s9 4 2.pg
A cell of some bacteria divides into two cells every 30 minutes.The initial population is 4 bacteria.
7. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/ur de 4 16.pg
Find the function satisfying the differential equation
(a) Find the size of the population after t hours
f 0 (t) − f (t) = 5t
and the condition f (2) = −6.
f (t) =
.
y(t) =
(function of t)
8. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/ur de 4 9.pg
A. Let g(t) be the solution of the initial value problem
(b) Find the size of the population after 3 hours.
y(3) =
7t
(c) When will the population reach 12?
with g(1) = 1.
Find g(t).
g(t) =
T=
1
dy
+ y = 0, t > 0,
dt
.
B. Let f (t) be the solution of the initial value problem
12.
with f (0) = 0.
Find f (t).
f (t) =
C. Find a constant c so that
(1
pt)
Library/Rochester/setDiffEQ4Linear1stOrder-
/ur de 4 13.pg
dy
7t + y = t 3
dt
Solve the initial value problem
8(sin(t)
.
dy
+ (cost)y) = (cos(t))(sin(t))6 ,
dt
for 0 < t < π and y(π/2) = 9.
y=
k(t) = f (t) + cg(t)
.
13. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/ur de 4 8.pg
Solve the initial value problem
solves the differential equation in part B and k(1) = 8.
.
c=
dy
+ 2y = 45 sin(t) + 30 cos(t)
dt
9. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/ur de 4 7.pg
Solve the initial value problem
with y(0) = 6.
y=
.
dy
8(t + 1) − 7y = 7t,
dt
for t > −1 with y(0) = 15.
y=
10.
(1
pt)
(Show the student hint after 4 attempts: )
Reminder: To find the anti-derivative of eu sin(u), the trick is to
do integration by parts twice.
.
Library/Rochester/setDiffEQ4Linear1stOrder-
/ur de 4 18.pg
14. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/ur de 4 3.pg
Find the function satisfying the differential equation
A Bernoulli differential equation is one of the form
dy
+ P(x)y = Q(x)yn .
dx
y0 − 3y = 2e4t
and y(0) = 4.
Observe that, if n = 0 or 1, the Bernoulli equation is linear.
For other values of n, the substitution u = y1−n transforms the
Bernoulli equation into the linear equation
y=
.
15. (1 pt) Library/Rochester/setDiffEQ2DirectionFields/ur de 2 5.pg
Use Euler’s method with step size 0.25 to compute the approximate y-values y1 , y2 , y3 , and y4 of the solution of the initial-value
problem
y0 = 2 − 3x + 4y, y(1) = −3.
du
+ (1 − n)P(x)u = (1 − n)Q(x).
dx
Use an appropriate substitution to solve the equation
y1 =
y2 =
y3 =
y4 =
xy0 + y = 2xy2 ,
and find the solution that satisfies y(1) = 3.
y(x) =
16. (1 pt) Library/UMN/calculusStewartET/s 9 2 21.pg
Use Euler’s method with step size 0.5 to compute the approximate y-values y1 , y2 , y3 , and y4 of the solution of the initial-value
problem
y0 = y − 3x, y(1) = 0.
.
11. (1 pt) Library/Rochester/setDiffEQ4Linear1stOrder/ur de 4 6.pg
Solve the following initial value problem:
dy
+ 0.7ty = 7t
dt
with y(0) = 2.
y=
,
,
,
.
1. y1 =
4. y4 =
.
2
2. y2 =
3. y3 =
18. (1 pt) Library/UMN/calculusStewartET/s 9 2 24.pg
Use Euler’s method with the given step size to estimate y(1.4)
where y(x) is the solution of the initial-value problem
17. (1 pt) Library/UMN/calculusStewartET/s 9 2 22.pg
Use Euler’s method with step size 0.2 to compute the approximate y-values y1 , y2 , y3 , y4 , and y5 of the solution of the initialvalue problem
y0 = 2 − xy, y(0) = 0.
1. y1 =
4. y4 =
2. y2 =
5. y5 =
y0 = x − xy,
y(1) = 3.
1. Estimate y(1.4) with a step size h = 0.2.
Answer: y(1.4) ≈
2. Estimate y(1.4) with a step size h = 0.1.
Answer: y(1.4) ≈
3. y3 =
c
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