The Volterra integral equation:
x(t) - xo = ∫[to, t] y(s) ds
y(t) - x₁ = ∫[to, t] f(s, x(s)) ds
(a) To write the initial value problem dx = f(t, x), x(to) = xo dt as an integral equation, we can use the integral representation of the solution.
Let's consider the integral equation:
x(t) = xo + ∫[to, t] f(s, x(s)) ds
This integral equation is of the Volterra type, as it involves a definite integral and depends on the function x(s) evaluated over the interval [to, t].
(b) To transform the second-order initial value problem d²x/dt² = f(t, x), x(to) = xo, x'(to) = x₁ into a Volterra integral equation, we can introduce a new variable, y(t), defined as the derivative of x(t):
y(t) = dx/dt
Now, we can rewrite the second-order differential equation as a system of first-order differential equations:
dx/dt = y(t)
dy/dt = f(t, x)
To represent this system as an integral equation, we'll integrate both equations from to to t.
This yields:
x(t) - xo = ∫[to, t] y(s) ds
y(t) - x₁ = ∫[to, t] f(s, x(s)) ds
Combining these two equations, we obtain the Volterra integral equation:
x(t) - xo = ∫[to, t] y(s) ds
y(t) - x₁ = ∫[to, t] f(s, x(s)) ds
This integral equation is also of the Volterra type, involving definite integrals and depending on the functions x(s) and y(s) evaluated over the interval [to, t].
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4x - 2y can be simplified as ____________
Answer:
4x - 2y can be simplified as 2(2x - y)
Consider the sequence of functions fr : [0,1] → R defined recursively by fo(x) = 1, fn(x) = Vxfn-1(x), n > 1. Prove that the sequence converges on [0, 1] and that the convergence is uniform.
For the sequence of functions fr : [0,1] → R defined recursively by fo(x) = 1, fn(x) = Vxfn-1(x), n > 1, the sequence converges on [0, 1] and the convergence is uniform.
First, we need to show that the sequence is pointwise convergent. To do that, we take an arbitrary x ∈ [0,1] and use induction on n to prove that fn(x) is convergent.
Let A = sup{f1(x), 1}
Since f1(x) = √x ≤ 1, there exists a decreasing sequence (an) such that f1(x) ≤ an ≤ 1 for all n.
Noting that an → √A (since an is decreasing and bounded below), we have:
fn(x) = √x*fn-1(x) ≤ √A * fn-1(x)
Now, by induction, we have:
f2(x) ≤ √A * f1(x) ≤ √A * a1 ≤ √Afn(x) ≤ √A * fn-1(x) ≤ √A * an-1 ≤ A for all n.
So, by the squeeze theorem, fn(x) is convergent and since this holds for all x, the sequence is pointwise convergent.
We need to show that the sequence is uniformly convergent on [0,1].
Let ε > 0 be arbitrary and let N be such that |an - A| < ε/2 for all n > N.
Let M be such that 1/M < ε/2.
We want to show that |fn(x) - A| < ε for all x ∈ [0,1] and all n > N.
If n ≤ N, then we can write:
|fn(x) - A| ≤ |f
n(x) - an| + |an - A| < ε for all x ∈ [0,1].
So, assume n > N. Then we have:
|fn(x) - A| = |fn(x) - an + an - A| ≤ |fn(x) - an| + |an - A| < ε/2 + ε/2 = ε for all x ∈ [0,1] by the definition of N and M.
Therefore, the sequence is uniformly convergent on [0,1].
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Find the equation of the line pls helppppppp
Answer:
y=2x+4
Step-by-step explanation:
y=2x+4
Answer:
y = 2x + 4
Step-by-step explanation:
The formula for the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. The y-intercept is where the line passes through the y-axis, which would be (0,4), which is positive. To find the slope, find where the line passes through a point, and then count how many units there are till the next point. For this example, you could start from the y-intercept, (0, 4) and to get to the next point, you will go up 2 units, and the right 1 unit. This means that the slope is 2/1 (because it's rise over run) which is the same as 2. So just plug it into the equation, and you get y = 2x + 4.
I hope this helped! :)
The last time you bough pizza, 3 pizzdis was enough for 7 people. At that rate, how
many pizzas should you buy for a party for 35 people.
Answer:
15 pizzas.
Step-by-step explanation:
The rate is 3 pizzas for every 7 people. If there are 35 people at the party (7 times 5), the amount of pizzas you should buy is 15 pizzas (3 times 5). Hope I explained that well (^^'')
When rolling a die, calculate the probability as a fraction of rolling a 3. The probability is
Answer:
1/6
Step-by-step explanation:
there is only one side in which there is 3 out of total six sides so 1/6
I'll give you the brainliest
Answer:
>
Step-by-step explanation:
sqrt{17} > sqrt{11}
Answer: >
Step-by-step explanation: the square root of 17 is going to be larger than the square root of 11. since you add 3 to both equations, that is all you have to worry about.
Please help! question : “ Give 3 possible solutions to x>2.”
Answer:
26 , 10 , 3
Step-by-step explanation:
Any number larger than 2 will work.
You place two wooden support beams under a shelf, as shown. Find the value of x
Answer:
17x
Step-by-step explanation:
so first 2x is 2 x x = 2x
then +10 =12x then 5 + x = 5x so 5x + 12x = 17x
Help tell me about scatter plots please!!!
Answer: i don’t know what that is
Step-by-step explanation:
If the measure of arc KM is 112°, find the measure of arc LM
Move point C and observe how the
angle measures change.
When AC changes,
m_ABC and mŁADC
vand are
to each other.
Answer:
[tex]\angle ABC[/tex] and [tex]\angle ADC[/tex] will change
[tex]\angle ABC[/tex] equal [tex]\angle ADC[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Required
What happens when AC changes
(a) [tex]\angle ABC[/tex] and [tex]\angle ADC[/tex] are formed from points A and C. So, a change in AC will result to a change in the measures of [tex]\angle ABC[/tex] and [tex]\angle ADC[/tex].
(b) [tex]\angle ABC[/tex] and [tex]\angle ADC[/tex] are in the same segment, meaning that they will always have the same measure
Suppose a regression was run between y and one x variable. The coefficient of the x variable (slope coefficient) was -0.76. The associated standard error of the slope coefficient was 0.04, and p value of the slope coefficient was 0. The t stat of the slope coefficient is: ________
The t-statistic of the slope coefficient is -19 is the answer.
Suppose a regression was run between y and one x variable.
The coefficient of the x variable (slope coefficient) was -0.76.
The associated standard error of the slope coefficient was 0.04, and p value of the slope coefficient was 0.
The t stat of the slope coefficient is -19.
To find the t-statistic value, divide the slope coefficient by the standard error.
This is given as t = slope / SE.
We have the values of the slope and the SE in the problem.
Substitute these values to get the t-statistic.t = -0.76 / 0.04 = -19
From this calculation, we can see that the t-statistic of the slope coefficient is -19.
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What should be done to both sides of the equation in order to solve y - 4.5 = 12.2?
Given: What should be done to both sides of the equation in order to solve y - 4.5 = 12.2?
To Find: What should be done on both side of equation.
Solution:
Step1 : Add 4.5 both sides so that +4.5 gets eliminated from LHS ,
=> y-4.5+4.5=12.2+4.5
=> y = 16.7
Hence the value of y is 16.7.
To need to solve the equation we need to add 4.5 on both sides and the value of y will be 16.7.
What is the equation?There are many different ways to define an equation. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of 2 mathematical expressions.
In other words, the equation must be constrained with some constraints.
As per the given equation,
y - 4.5 = 12.2
Add 4.5 on both sides of the equation,
y - 4.5 + 4.5 = 12.2 + 4.5
y = 16.7
Hence "To need to solve the equation we need to add 4.5 on both sides and the value of y will be 16.7".
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Let W = {a + 2x + bx2 € Pz: a, b E R} with the standard operations in P2. Which of the following statements is true? W is not a subspace of P2 because 0 € W. W is a subspace of P2. a The above is true The above is true None of the mentioned O 1+xEW
The statement "W is a subspace of P2" is true, as W satisfies the conditions for being a subspace: closure under addition and scalar multiplication, and containing the zero vector.
The statement "W is a subspace of P2" is true.
To show that W is a subspace, we need to verify the three conditions for subspace:
W is closed under addition: For any two polynomials p(x) = a + 2x + bx^2 and q(x) = c + 2x + dx^2 in W, their sum p(x) + q(x) = (a + c) + 4x + (b + d)x^2 is also in W. Therefore, W is closed under addition.
W is closed under scalar multiplication: For any polynomial p(x) = a + 2x + bx^2 in W and any scalar k, the scalar multiple kp(x) = ka + 2kx + kbx^2 is also in W. Therefore, W is closed under scalar multiplication.
W contains the zero vector: The zero polynomial 0 = 0 + 0x + 0x^2 is in W since a = b = 0. Therefore, W contains the zero vector.
Since W satisfies all three conditions for subspace, it is indeed a subspace of P2.
Hence, the statement "W is a subspace of P2" is true.
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Nia increased her class average by 3%. She now has a 92% in her math class.
Nia's original class average was 89%. After increasing it by 3%, she now has a 92% in her math class.
1. Let's assume Nia's original class average was X.
2. She increased her class average by 3%, which can be represented as X + (3/100)X.
3. According to the question, X + (3/100)X = 92.
4. Simplifying the equation, we have (103/100)X = 92.
5. To find X, we divide both sides of the equation by (103/100): X = (92 * 100) / 103.
6. Calculating the value, X ≈ 89.32.
7. Therefore, Nia's original class average was approximately 89.32%.
8. After increasing it by 3%, her new class average is 89.32% + (3/100) * 89.32% ≈ 92%.
Note: The values have been rounded for simplicity.
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⚠️⚠️⚠️help⚠️⚠️⚠️⚠️⚠️
Answer:
2
Step-by-step explanation:
The Intial value, also know as the y-intercept, is where the line crosses line y.
Therefore, looking at the graph, your initial value is 2.
Flip a biased coin twice. Assume that P[Head) = p. Denote "l" as head, and "0" as tail. Let X be the maximum of the two numbers, and let y be the minimum of the two numbers. (a) Find and sketch the joint PMF px,y(x,y). (b) Find the marginal PMF px) and py(y). (c) Find the conditional PMF Pxy(x|y).
The joint PMF, marginal PMFs, and conditional PMF can be determined for the maximum and minimum outcomes when flipping a biased coin twice. The joint PMF describes the probabilities of different combinations of maximum and minimum outcomes, while the marginal PMFs represent the probabilities of individual outcomes.
The conditional PMF shows the probability of one outcome given another. Detailed calculations and explanations are required to provide a complete answer.
(a) To find the joint PMF px,y(x,y), we need to consider all possible outcomes of flipping the coin twice. Since X represents the maximum outcome and Y represents the minimum outcome, we can determine the probabilities for each combination of X and Y. For example, if X = 1 and Y = 0, it means that the first flip was a tail and the second flip was a head. The joint PMF will assign probabilities to all possible combinations of X and Y.
(b) The marginal PMFs px(x) and py(y) represent the probabilities of individual outcomes for X and Y, respectively. To find px(x), we sum up the probabilities of all combinations where X takes the value x. Similarly, to find py(y), we sum up the probabilities of all combinations where Y takes the value y.
(c) The conditional PMF Pxy(x|y) provides the probability of X taking a certain value given that Y has a specific value. It can be obtained by dividing the joint PMF px,y(x,y) by the marginal PMF py(y) for each y value.
To provide a more detailed answer with calculations and sketches, specific values for p, the probability of a head, are needed.
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What would be the absolute value of Point R shown on the number line?
Answer:
|-2| = 2
Step-by-step explanation:
complete the proof given: AB x BE =CB x BD prove: triangle ABC triangle DBE
The complete proofs with all the reasons and statements; have been enumerated below
Triangle Proofs
The image showing the given triangle is missing and so i have attached it.
From the attached image, we can deduce that;
Statement 1; AB * BE = CB * BD
Reason 1; Given
Statement 2; CB/BE = AB/BD
Reason 2; Property of Cross Multiplication
Statement 3; ∠ABC ≅ ∠DBE
Reason 3; Vertical Angles
Statement 4; ΔABC ~ ΔDBE
Reason 4; SAS property of similarity
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Do,k= (0,2) (0,6) the scale factor is
3
0
1/3
Answer:
Step-by-step explanation:
3
The temperature at the beginning of the day was 6 degrees F. The temperature dropped 9 degrees F by the end of the day. What was the temperature at the end of the day?
Temperature is an average measure of the kinetic energy of particles of matter.
Temperature can be in negative degrees.
The temperature at the end of the day is -3°F.
What is temperature?Temperature is an average measure of the kinetic energy of particles of matter.
The Celsius scale is generally used for measuring most temperatures.
The Fahrenheit (°F) temperature scale is also used.
We have,
The temperature at the beginning of the day = 6°F
The temperature drop by the end of the day = 9°F
Temperature can also be in negative degrees.
The temperature at the end of the day:
= 6°F - 9°F
= -3°F
Thus the temperature at the end of the day is -3°F.
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The National Health Survey uses household interviews to describe the health-related habits of U.S. adults. From these interviews they estimate population parameters associated with behaviors such as alcohol consumption, cigarette smoking, and hours of sleep for all U.S. adults. In the 2005-2007 report, they estimated that 30% of all current smokers started smoking before the age of 16. We randomly select a sample of 100 smokers and calculate the proportion who started smoking before the age of 16.
Answer: hello your question is incomplete below is the missing part
Based on your simulation, what is the average amount of error you expect to see in the sample proportions in this situation?
answer : 0.0458
Step-by-step explanation:
In 2005 - 2007 : 30% of all current smokers started smoking before the age of 16
n ( sample size ) = 100 smokers
The average amount of error expected can be calculated as
[tex]\sqrt{pq/n}[/tex] = [tex]\sqrt{0.3*0.7/100}[/tex] = 0.0458
where : p = 0.3
q = 0.7
n = 100
Convert the time. Enter your answers in the boxes.
324 minutes =
hours,
minutes
Answer:
5 hours and 24 minutes
Step-by-step explanation:
sorry if im wrong
Answer:
5.4 hours
Step-by-step explanation:
I'm not 100 percent sure
hope this helped ;)
An electric frying pan was on sale for 20 percent off. The sale price was $36. What was the original price?
Answer: $45
Step-by-step explanation:
Let the original price be x.
Since there was a discount of 20%, it means the person bought the frying pan at (100% - 20%) = 80% of the original price. This can then be formed into an equation as:
80% × x = $36
0.8 × x = $36
0.8x = $36
x = $36/0.8
x = $45
The original price was $45
I need help with this math promblem will anybody help me
Answer:
d = 83
Step-by-step explanation:
Answer:
d=83°
Step-by-step explanation:
the sum of the inner angles is equal to 360°
so
127+67+d+d=360
194+2d=360
2d=166
d=83°
Graph the line.
y+4= -1/3(x+5)
Answer:
graph is shown
Step-by-step explanation:
slope: -1/3
y intercept: (0,-17/3)
Can someone help me please
Answer:
1: x = 6
2: There is no solution. Sorry. (because the ending equation will be 0 = 7)
3: I can't see the other part of the equation, I need to see what the fraction is in order to solve it. If you can retake the picture, I can edit my response :)
Step-by-step explanation:
#1: 3(x-1) is pretty much just 3x-3. Simple.
Now, your equation becomes 3x-3 = 2x+9.
Arrange them so that the like terms are together.
Since you will be taking the 2x from 2x+9 over to the other side, it becomes 3x-2x-3 = 9.
Now, bring the 3 over to the side with the 9.
x = 6.
#2: Same as before, -2(x+1) is (-2x) - 2.
So, -2x-2 = -2x+5.
As I said before, you need to group the like terms together.
-2x and -2x go together, and -2 and 5 go together.
-2x+2x=5+2
0=7.
There is no solution.
#3: Like the other two, 4(x-1) is 4x-4. As for the other part, I cannot see the fraction before (x-8).
are these proportional?
3/2 and 18/8
Answer:
For two fractions to be proportional, they must be equivalent. 32=1.5 and 188=2.25 , so these fractions are not proportional.
Step-by-step explanation:
Please help due two days ago helpp me please Which measure of center is the most appropriate for the data in Table 1 (Height) in Task 1? Give a reason for your answer. calculate the value of the most appropriate center value of the heights.
Answer:
I think the correct answer is....... The measure of center for table 1 is 61, 62, 58, and 57 inches. The most kids had the same number of inches in this category.
I'm not 100% sure so try at your own risk
Have a great day!
The center of measures of table 1 height (task1) are Mean of heights of siblings are 58.29. Median of the height of 24 siblings are 61.5. Mode of the height 24 siblings 57,58,61,62 are the heights in inches.
What is Measures of center?
Measures of center of data set is also "a way of describing data set. The two most widely used measures of the center is mean and median".
What is mean?
Mean is the process of "adding all values and then divided by total number of values".
What is median?
Median is the "process of arranging data value in ascending or descending order and then divided data set into two if 'n' is odd then middle point is the median or if 'n' is even then sum of two mid value and then divided by two".
What is Mode?Mode can be one, "two or more in the given data set. The number or data point which have more frequency is taken as mode".
According to the question,
Measures of center is the most appropriate for the data in Table 1 (Height) in Task 1.
Heights in inches Frequency
55 1
57 3
58 3
59 2
60 1
61 3
62 3
63 2
64 2
65 2
66 1
67 1
Total number of frequency = N = 1+3++3+2+1+3+3+2+2+2+1+1 =24. Number of siblings n =12
Mean = ∑(Height of inches × Frequency of height)/Total number of frequency.
= [tex]\frac{(55*1+57*3+58*3+59*2+60*1+61*3+62*3+63*2+64*2+65*1+66*1+67*1)}{24}[/tex]
= [tex]\frac{1399}{24}[/tex]
= 58.29.
Mean of heights of siblings are 58.29
Median of the height of 24 siblings are sum of 61 and 62 divided by 2.
= [tex]\frac{61+62}{2}[/tex] [since 'n' is odd]
=61.5.
Median of the height of 24 siblings are 61.5.
Mode of the height 24 siblings 57,58,61,62 are the heights in inches.
Hence, the center of measures of table 1 height (task1) are Mean of heights of siblings are 58.29. Median of the height of 24 siblings are 61.5. Mode of the height 24 siblings 57,58,61,62 are the heights in inches.
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PLEASE ANSWER THIS ASAP I WILL MARK YOU THE BRAINLIEST
SHOW YOUR WORK!!!
Calculate the volume of the following three-dimensional object
Answer: 6773.27
Step-by-step explanation: volume formula for the cone is pie(radius)^2(height/3) just plug in your information into the formula and viola