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Homework answers / question archive / 1)Consider the graph 2 − 2 − 4 2 4 6 − 2 x y Using the above graph of y(x), choose the equation whose solution satisfies the initial condition y(0) = 3
1)Consider the graph
2 
− 
2 
− 
4 
2 
4 
6 
− 
2 
x 
y 
Using the above graph of y(x), choose the equation whose solution satisfies the initial condition y(0) = 3.5.
2. From this, we can see that the given graph is indeed a solution to the differential equation y^{′ }= y − x.
Consider the graph
2 
− 
2 
− 
4 
2 
4 
6 
− 
2 
x 
y 
Using the above graph of y(x), choose the equation whose solution is graphed and satisfies the initial condition y(0) = 1.5.
4. 
− 
2 
2 
− 
2 
2 
x 
y 
Use the direction field of the differential equation y^{′ }= y^{3 }to sketch a solution curve that passes through the point (0, 1).
− 
2 
2 
− 
2 
2 
x 
y 
1. 
− 
2 
2 
− 
2 
2 
x 
y 

2. 
− 
2 
2 
− 
2 
2 
x 
y 
6. 
− 
2 
2 
− 
2 
2 
x 
y 
:
A solution y(t) of the differential equation
y^{′ }= y^{3 }− 4y
satisfies the initial condition y(0) = 1. Find lim y(t). t→−∞
.
y^{′ }= 1 + 5x − 3y, y(1) = 2.
rect
In Euler’s Method, we predict the value of y_{i }using, and dx with the following equation yi = yi−1 + ^{y}_{i}^{′}_{−}1 dx.
Recall that y^{′}(x,y) = 1 + 5x − 3y and y(1) = 2. Hence, in this problem 2) = 0 and dx = 0.5.
In order to visualize this better, let us use the following table to display our values
i 
x 
y 
y′ 
0 
1.0 
2.000 
0.000 
1 
1.5 
2.000 
2.500 
2 
2.0 
3.250 
1.250 
3 
2.5 
3.875 
1.875 
Notice in this table, y is computed in each row using the values for y and y^{′ }in the previous row while y^{′ }in each row is determined using y and x in the current row. Hence,
y_{1 }= 2.000, y_{2 }= 3.250, y_{3 }= 3.875 
.
.
.
,
find its general solution.
1.
2.
3.
4.
5.
3.
4.
5.
,
find the value of y_{0}(2).
,
find the general solution.
dy ^{x}^{+3}, y(−3) = 0,
2y= e dx
and y > 0, find the value of y_{0}(0).
,
Find its general solution.
1.
2. 3.
4.
18. Find the particular solution y_{1 }such that y_{1}(8) = 5.
1 ^{2}
9
1 ^{2}
3
4.
5.

83
9
83
9
113
3
143
3
.
dy ^{2}),
y= x(36 + y dx
which satisfies y(0) = 0, find the value of y_{0}(1).
1.
2.
3.
4.
5.
dy
+ 2xy + 4x = 0,
dx such that y(0) = 3, find the value of
1.
2.
3.
4.
5.
,
(i) First find the general solution.
1 ^{x}^{+1 }+ A)
7
5.
1 1 x+1 − 1)1/2
7
1
7
1.
1
7
25. For the differential equation
(i) first find its general solution.
1.
2.
3.
2.
3.
4.
5.
28. If y_{1 }satisfies the equations
dy ^{2 }lnx, y(1) = e, 2xy= 3 + 16x dx
determine the value of y_{1}(e).
29. If y_{1 }satisfies the equations
1
for x > , determine the value of y_{1}(e).
2
1
13
1
12
1
9
1
11
30. A differentiable function f has the property that
holds for all x, y in the domain of f. It is
known also that
.
(i) Use the definition of the derivative of f to determine f^{′}(x) in terms of f(x).
31. (ii) By solving the differential equation in Part
(i) for f, find the value of
1.
2.
3.
4.
5.
32. An initial deposit of $P is made into an account that earns 5% interest compounded continuously. Money is then withdrawn at a constant rate of $3000 per year.
Set up the differential equation for the amount A = A(t) (in thousands of dollars) in the account after t years.
dA
33. Solve the differential equation in part 1.
1.
2.
3.
4.
5.
A 
( 
t 
)=60+ 

P 
1000 
− 
60 

e 
0 
. 
05 
t 
. 
34. If the account balance becomes zero after 10 years, what was the amount of the initial deposit $P? (This value of P is often called the Present Value of the regular withdrawals $3000 over a period of n years.)
35. In a West Texas school district the school year began on August 1 and lasted until May
31. On August 1 a Soft Drink company installed soda machines in the school cafeteria. It found that after t months the machines generated income at a rate of
dollars per month. Find the total income produced during the first semester ending on December 31.
36. A flea collar for dogs contains an active ingredient that evaporates at a rate proportional to the amount still present. Half of the ingredient evaporates in the first 18 days after the collar is removed from the protective packaging.
If the collar becomes ineffective after 60% of the active ingredient has evaporated, how many days does the collar remain effective?
37. A population is modeled by the differential equation.
For what values of P is the population increasing?
38. A common inhabitant of human intestines is the bacterium Escherichiacoli. A cell of this bacterium in a nutrientbroth medium divides into two cells every 20 minutes, and its initial population is 65 cells.
Find the number of cells after 4 hours.
39. Experiments show that if the chemical reaction
N
takes place at 45^{?}C, the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows
.
How long will the reaction take to reduce the concentration of N_{2}O_{5 }to 70% of its original value?
40. Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Suppose that a roast turkey is taken from an oven when its temperature has reached 160^{?}F and is placed on a table in a room where the temperature is 70^{?}F.
If u(t) is the temperature of the turkey after t minutes, then Newton’s Law of Cooling implies that
.
This could be solved as a separable differential equation. Another method is to make the change of variable y = u − 70^{?}F. ?
If the temperature of the turkey is 130 F after half an hour, what is the temperature after 20 min?
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