Solve the homogeneous equation
[tex]y'' + 3y' + 2y = 0[/tex]
Its characteristic equation is
[tex]r^2 + 3r + 2 = (r + 1) (r + 2) = 0[/tex]
with roots at [tex]r=-1[/tex] and [tex]r=-2[/tex], hence the characteristic solution is
[tex]y_c = C_1 e^{-x} + C_2 e^{-2x}[/tex]
For the nonhomogeneous equation, I'll use variation of parameters. We're looking for a solution of the form
[tex]y = u_1 y_1 + u_2 y_2[/tex]
to the equation
[tex]y'' + a(x) y'' + b(x) y = f(x)[/tex]
such that
[tex]\displaystyle u_1 = - \int \frac{y_2f(x)}{W(y_1,y_2)} \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{y_1 f(x)}{W(y_1,y_2)} \, dx[/tex]
The Wronskian [tex]W(y_1,y_2)[/tex] of the two fundamental solutions [tex]y_1=e^{-x}[/tex] and [tex]y_2=e^{-2x}[/tex] is
[tex]W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ {y_1}' & {y_2}' \end{vmatrix} = -e^{-3x}[/tex]
Then we have
[tex]\displaystyle u_1 = - \int \frac{e^{-2x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = 4 \int e^{2x} \cos(3x) \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{e^{-x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = -4 \int e^{3x} \cos(3x) \, dx[/tex]
Recall Euler's identity,
[tex]e^{(a+bi)t} = e^{at} (\cos(bt) + i \sin(bt))[/tex]
Then we have the general antiderivative
[tex]\displaystyle \int e^{(a+bi)t} \, dt = \frac1{a+bi} e^{(a+bi)t} + C = \frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C[/tex]
Taking the real parts of both sides, we have
[tex]\displaystyle \mathrm{Re}\left\{\int e^{(a+bi)t} \, dt \right\} = \mathrm{Re}\left\{\frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C\right\} \\\\ \int\,\mathrm{Re}\left\{e^{(a+bi)t}\right\} \, dt = \frac{e^{at}}{a^2+b^2} \mathrm{Re}\left\{(a-bi)(\cos(bt) + i \sin(bt))\right\} + C \\\\ \int e^{at} \cos(bt) \, dt = \frac{e^{at}}{a^2+b^2} (a\cos(bt)+b\sin(bt)) + C[/tex]
so that
[tex]\displaystyle u_1 = 4 \int e^{2x} \cos(3x) \, dx = \frac{4e^{2x}}{13} (2\cos(3x) + 3 \sin(3x))[/tex]
and
[tex]\displaystyle u_2 = -4 \int e^{3x} \cos(3x) \, dx = -\frac{2e^{3x}}3 (\cos(3x) + \sin(3x))[/tex]
We've found
[tex]y = u_1 y_1 + u_2 y_2[/tex]
[tex]\displaystyle y = \frac{4e^x}{13} (2\cos(3x) + 3 \sin(3x)) - \frac{2e^x}3 (\cos(3x) + \sin(3x))[/tex]
[tex]\displaystyle y = \frac2{39} e^x (5\sin(3x) - \cos(3x))[/tex]
Then the general solution to the differential equation is
[tex]\boxed{y(x) = C_1 e^{-x} + C_2 e^{-2x} + \frac2{39} e^x (5\sin(3x) - \cos(3x))}[/tex]
The perimeter of the rectangle below is 84 units. Find the length of side VW.
Write your answer without variables.
Answer:
18.
Step-by-step explanation:
1) formula of the perimeter given in the condition is:
p=2(VW+XV) or p=2(VY+XY).
Note, VW=XY and XV=VY;
2) if to substitute p=84, VW=2z+2 and XV=3z into the formula above, then
2(2z+2+3z)=84, ⇔ 5z+2=42; ⇒ z=8.
3) if z=8, then VW=2z+2=2*8+2=18.
A picture frame is 20 cm long and 30 cm wide. This includes a pink border that is 5
cm wide surrounding the picture itself, as shown below. What is the area of this pink
border in square centimeters?
20 cm
30 cm
The area of the rectangular pink border is of 400 cm².
What is the area of a rectangle?The area of a rectangle of dimensions l and w is given by the multiplication of the dimensions, as follows:
A = lw.
Hence, the total area is given by:
A = 30 x 20 = 600 cm².
The area of the white region is given by:
A = (30 - 2 x 5) x (20 - 2 x 5) = 20 x 100 = 200 cm².
Hence the area of the pink border is:
600 cm² - 200 cm² = 400 cm².
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O-13
13
O-5
O 5
Simplify 9-(-4)=
Answer:
B. 13
Step-by-step explanation:
[tex]Rule:a-\left(-b\right)=a+b\\=9-\left(-4\right)=9+4\\=9+4\\=13[/tex]
Dependent events can best be described as
two or more events that have the same probabilities.
two or more events that do not have the same probabilities.
a group of events whose later outcomes are not affected by previous outcomes.
a group of events whose later outcomes are affected by previous outcomes.
.
Option D is the right choice. Dependent events can best be described as group of events whose later outcomes are affected by previous outcomes.
We are aware of two instances. If the occurrence of one result influences the occurrence of the other outcome, then events A and B are said to be dependent.
For instance,
If we pull the first marble without replacing it from a bag containing 5 blue marbles, 2 red marbles, and 3 green marbles.
The sketching of a blue marble first, followed by a green marble, is an instance of a dependent event.
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Answer:
d on edge
Step-by-step explanation:
Last week, a chocolate shop sold 9 ounces of white chocolate. It sold 9 9/10 times as much milk chocolate as white chocolate. How many ounces of milk chocolate did the shop sell?
please answer asap
Solving the Question
We're given:
9 ounces of white chocolate sold[tex]9\dfrac{9}{10}[/tex] times as much milk chocolate as white chocolate soldIf the shop sold " [tex]9\dfrac{9}{10}[/tex] times as much milk chocolate as white chocolate", we must multiply [tex]9\dfrac{9}{10}[/tex] by the amount of white chocolate sold to find the amount of milk chocolate sold.
Multiply [tex]9\dfrac{9}{10}[/tex] by 9 ounces:
[tex]9\dfrac{9}{10}\times9[/tex]
Convert the fraction into an improper fraction:
[tex]=\dfrac{99}{10}\times9[/tex]
Multiply:
[tex]=\dfrac{891}{10}[/tex]
AnswerThe shop sold [tex]\dfrac{891}{10}[/tex] ounces of milk chocolate.
Question 3
1 pts
A survey was conducted by visiting a student parking lot to estimate the proportion of
cars that were red. Which of the following is NOT correct?
A. If another sample of cars was chosen, it is likely that a different proportion of cars that are red
would be obtained.
B. The sample would be representative of the population if 100 cars were chosen regardless if
randomization was used or not.
C. Even though a random sample was taken from cars in the parking lot, the sample may not be
representative of the cars driven by students because the decision to park in that particular lot is
self-selected.
D. If the sampled stall was empty, we can simply choose another stall, at random, to take its place
because it is not likely that the stall being vacant is related to a car being red.
The sample would be representative of the population if 100 cars were chosen regardless if randomization was used or not; this statement is not correct;
A survey was carried out by visiting a student parking lot to estimate the number of red cars.
If a different sample of cars was picked, the proportion of red cars would most likely be different. This statement is correct.
The sample would be representative of the population if 100 cars were chosen regardless if randomization was used or not. This statement is not correct because you select only 100 cars to represent whole population that too without randomization;
Despite the fact that a random sample was obtained from cars in the parking lot, the sample may not be representative.
because the decision to park in that particular lot is typical of the cars driven by students self-selected.
If the sampled stall was empty, we can simply choose another stall at random to take its place because the stall being unoccupied is unlikely to be associated to a red car.
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____________________________________________________________
______________________________
a 42 inch string is to be cut so that one piece is 7 inches longer than the other. what is the length of the shorter piece?
Answer:
17.5 inches
Step-by-step explanation:
Let the length of the shorter piece be x
Shorter piece + Longer piece = Whole string
x + (x + 7) = 42
x + x + 7 = 42
Both sides minus 7 to remove the 7 from the left side of the equation.
2x + 7 - 7 = 42 - 7
2x = 35
x = 17.5 inches
Hope this helped! ^^
S= 1/2( a + b + c) solve for b
The solution of the equation for b is b = 2s - a - c
How to solve for b in the equation?The equation is given as
s= 1/2(a + b + c)
Multiply through by 2
So, we have
2s = a + b + c
Subtract a and c from both sides of the equation
b = 2s - a - c
Hence, the solution of the equation for b is b = 2s - a - c
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Mark all of the statements that are true.
A. The domain for this function is the set {-5).
B. All real numbers are in the range of this function.
C. The domain for this function is all real numbers.
D. The range for this function is the set {-5).
E. This graph is not a function because the value of y is the same for every value of x
Option E:
This graph is not a function because the value of y is the same for every value of x
Solution :
Domain:
The domain of a function is the set of its possible inputs, i.e., the set of input values where for which the function is defined. In the function machine metaphor, the domain is the set of objects that the machine will accept as inputs.
For example, when we use the function notation f:R→R, we mean that f is a function from the real numbers to the real numbers. In other words, the domain of f is the set of real number R (and its set of possible outputs or codomain is also the set of real numbers R).
relation represents y as a function of x
1) is NOT a function of if there are two or more points with the same − , but
− .
2) is a function of if each has a different
There is a technique called the vertical line test that is often used to determine if a graph represents y
as a function of x
In this question only a horizontal line is present.
Option E:
This graph is not a function because the value of y is the same for every value of x
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What is the unit rate?
For the given proportional relationship, it is found that the unit rate is of 1.5 inches per second, and the meaning of the points is given as follows:
(0,0): 0 inches in 0 seconds.(1, 1.5): 1.5 inches in 1 second.(3, 4.5): 4.5 inches in 3 seconds.What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:
y = kx
In which k is the constant of proportionality, also called the unit rate.
For this problem, the unit rate is given as follows:
k = 4.5/3 = 1.5/1 = 1.5 inch per second.
The meaning of the points are given by:
(0,0): 0 inches in 0 seconds.(1, 1.5): 1.5 inches in 1 second.(3, 4.5): 4.5 inches in 3 seconds.More can be learned about proportional relationships at https://brainly.com/question/10424180
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a repair bill for your ebike is $188. the parts costs $73 dollars and the cost of the labor is $46 per hour.
a. write an equation that represents this situation.
b. how many hours dud the repair take?
Answer:
188 = 46x + 73
It took 2 1/2 or 2.5 hours to complete.
Step-by-step explanation:
There is a one time cost of 73 for the parts, but the hours will be 46 per hour. I know the total cost, so we need to solve for x which is the hours.
188 = 46x + 73 Subtract 73 from both sides of the equation
115 = 46x Divide both sides of the equation by 46
2.5 = x
The recommended daily allowance of niacin for adults is 20 mg. One serving of a certain breakfast cereal provides 4 mg of niacin. What percent of the recommended daily allowance of niacin does one serving of this cereal provide?
Answer:
20%
Step-by-step explanation:
4/20 * 100% = 20%
Which mixed number is equivalent to the improper fraction?
2/
OA. 22
O B. 22
7
O C. 11/1/2
O D. 32
Answer:
answer should be A. 2⅖
Step-by-step explanation:
you can find out by turning it into an improper fraction: =5×2+2/5
=10+2/5
=12/5
For the equation, find three ordered pair solutions by completing the table. Then use any two of the ordered pairs to graph the equation.
For the linear function y = 0.5x. three ordered pairs are given as follows:
x = 0, y = 0.x = 2, y = 1.x = -2, y = -1.Using points (-2,-1) and (2,1), the graph of the function is given at the end of the problem.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.In this problem, the function is:
y = 0.5x.
Hence three pairs of points are given as follows, replacing the value of x in the equation to find y.
x = 0, y = 0.x = 2, y = 1.x = -2, y = -1.The graph of the function is given at the end of the answer.
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telefonika a mobile phone shop has dicovered that 35% of all Iphones XS screens sold last month are detective. It has contacted the owners of these phone and ask them to return their phones to be checked and their screen replaced for them. Telefonika sold seven of these phones and has two of the new screens in stock. what is the probability that telefonika will need to order more new screens?
Answer: he will need to order 1 more new screens
Step-by-step explanation:
a mobile phone shop has discovered that 35% of all I phones XS screens sold last month are defective.
so probability of defective screen is
x=35/100 (let)
now since he sold 7 phones
so number of defective screen can be
=x*7
=2.45
since defective screen cant be in decimal
so
taking into consideration that last screen have a chance of getting defective
so he would need 3 screen
he has 2 in stock
so he will need to order 1 more new screens
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solve: let f(x)=2x-1 and g(x)=x^2-4. find g(f(x))
Work Shown:
g(x) = x^2 - 4
g(f(x)) = ( f(x) )^2 - 4
g(f(x)) = ( 2x-1 )^2 - 4
g(f(x)) = (2x-1)(2x-1) - 4
g(f(x)) = 4x^2-2x-2x+1 - 4
g(f(x)) = 4x^2 - 4x - 3
The histogram shows the number of miles driven by a sample of automobiles in New York City. What is the class width
The class width of the histogram is 5000.
What is class width?It should be noted that the class width is the distance that can be found between the lower limits of consecutive classes.
The range is the difference between the maximum and minimum data entries.
class width of a histogram is given by the equation
Therefore, class width:
= maximum of a class - minimum of a class
In the given histogram, first-class start at 2500 and ends at 7500..Therefore considering first class,.minimum =2500 and maximum=7500.
Therefore, class width will be:
= 7500-2500
= 5000
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write the slope-intercept form of the equation for each line
New Jersey transit claims that it’s 8:00 AM train from Princeton to New York has probability 0.9 of arriving on time. I assume for now that this claim is true. Explain what probability 0.9 means in the setting
It means there's a 90 percent chance that 8:00 AM train from Princeton to New York will arrive on time, and 10 percent chance it’s 8:00 AM train from Princeton to New York will not arrive on time.
Probability refers to possibility. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution. Knowing the total number of outcomes is necessary before we can calculate the likelihood of a particular event happening.
It means there's a 90 percent chance that 8:00 AM train from Princeton to New York will arrive on time, and 10 percent chance it’s 8:00 AM train from Princeton to New York will not arrive on time.
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refer to the following equation:y=2x-1. complete the table below to find co-ordinates
Answer:
Step-by-step explanation:
y=2x1-
So this is a very simple question.
First, analyze the problem
It says y=2x-1
The answer is +1 or 1 I believe
Someone pls correct me If I'm wrong.
The results of a survey of students about whether they attended the dance are shown in the table. What is the best estimate of the probability that a randomly selected boy attended the dance?
a. 7/20
b. 7/15
c. 7/9
d. 7/8
Answer:
7/9 the answer is C
Step-by-step explanation:
78/99 simplified to the smallest form
Answer:
The simplest form of
78/99
is
26/33
.
HELP ASAP!!!
Determine the direction in which the graph of the function f(x)=x^2+x-2 opens.
A.upward
B.downward
C.to the left
D.to the right
Answer:
A
Step-by-step explanation:
Since the leading coefficient of the function is greater than 0, then it opens upward.
whats the distance between points p1=(-3,5) P2=(4,2)
Step-by-step explanation:
First, find the vector.
= (4,2) - (-3,5)
= (4-(-3),2-5)
= (7,-3)
And then, find the distance.
= √(7² + (-3)²)
= √(49 + 9)
= √58 ✓
f(x) = 2x - 4
f(3) = [?]
Answer: f(3)=2
Step-by-step explanation:
f(x) = 2x - 4
f(3) =2(3) - 4
f(3) = 6-4
f(3)=2
Simplify negative 2 and 3 over 5 minus 6 and 2 over 7.
negative 12 and 1 over 2
negative 8 and 31 over 35
negative 8 and 6 over 35
12 and 6 over 35
The value of the expression is -8 31/35
How to simplify the given expression?The expression is given as:
negative 2 and 3 over 5 minus 6 and 2 over 7.
Rewrite properly as
- 2 3/5 - 6 2/7
Express as improper fraction
-13/5 - 44/7
Take the LCM
(-13 * 7 - 44 * 5)/35
Evaluate the difference of products
-311/35
Express as mixed fraction
-8 31/35
Hence, the value of the expression is -8 31/35
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Answer:
B
Step-by-step explanation:
Which choices are equivalent to the expression below check all that apply
Using exponential function, Correct answers are D and F. 8².8⁵ is equivalent to 8²⁺⁵, 8⁷
What is an exponential function?An exponential function is a mathematical function with the expression f (x) = ax. where x is a variable and an is a constant known as the function's base. The most common exponential-function base is the transcendental number e, or around 2.71828.
What traits do exponential functions possess?
When the base is bigger than 1, the exponential function has the following characteristics.
The point is traversed by the graph (0,1)The domain only contains true numbers.It is between y>0.The graph is going up.As x approaches negative infinity, the graph asymptotically approaches the x-axis.The value of 8².8⁵ is -
8²⁺⁵
8⁷,
Which is equivalent to 7 times 8.
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Suppose you have $5000 to invest. A certificate of deposit (CD) earns 6% annual interest, while bonds are much riskier,
earn 8% annual interest. You decide to invest $2000 in a (CD) and the balance in bonds. How much interest would you have
earned at the end of one year? How much interest would you have earned compounded annually after 2 years?
Get the first year interest. Also, calculate the second year interest then add it to the first to get total interest after 2 years.
Remember: Interest - Principal x Rate x Time
A. $360, $794.40
B. $120, $240
C. $360, $746.40
Answer:
b
Step-by-step explanation:
earned at the end of one year? How much interest would you have earned compounded annually after 2 years?
Get the first year interest. Also, calculate the second year interest then add it to the first to get total interest after 2 years.
Remember: Interest - Principal x Rate x Time
30 POINTS...PLEASE HELP ME ASAP
Look for a pattern in the table of values.
X -3 -2 -1 0 1 2
Y 32 16 8 4 2 1
A. Which model (linear, exponential, or quadratic) best describes the data?
B. Write an equation for the function that models the data.
wait a second...............