Question 2
Use the technique of Laplace transformation to solve the differential equation
d^2y/dx +y=0 dx
for the initial conditions
dy(0)/dx = 2, y(0) = 1
Answers
To use the Laplace transformation to solve the following differential equation, we will first apply the transformation to the problem and its initial conditions. F(s) denotes the Laplace transform of a function f(x) and is defined as: \(Lf(x) = F(s) = [0,] f(x)e(-sx)dx\)
When the Laplace transformation is applied to the given differential equation, we get:
\(Ld2y/dx2/dx2 + Ly = 0\) .
If we take the Laplace transform of each term, we get: \(s^2Y(s) = 0 - sy(0) - y'(0) + Y(s)\).
Dividing both sides by \((s^2 + 1),\), we obtain:
\(Y(s) = (s + 2) / (s^2 + 1)\).
Now, we can use the partial fraction decomposition to express Y(s) in terms of simpler fractions:
Y(s) = (s + 2) / (\(s^{2}\)+ 1) = A/(s - i) + B/(s + i) .
Multiplying through by (\(s^{2}\) + 1), we have:
s + 2 = A(s + i) + B(s - i).
Expanding and collecting like terms, we get:
s + 2 = (A + B)s + (Ai - Bi).
Comparing the coefficients of s on both sides, we have:
1 = A + B and 2 = Ai - Bi.
From the first equation, we can solve for B in terms of A:B = 1 - A Substituting B into the second equation, we have:
2 = Ai - (1 - A)i
2 = Ai - i + Ai
2 = 2Ai - i
From this equation, we can see that A = 1/2 and B = 1/2. Substituting the values of A and B back into the partial fraction decomposition, we have:
Y(s) = (1/2)/(s - i) + (1/2)/(s + i). Now, we can take the inverse Laplace transform of Y(s) to obtain the solution y(x) in the time domain. The inverse Laplace transform of 1/(s - i) is \(e^(ix).\)
As a result, the following is the solution to the given differential equation:\((1/2)e^(ix) + (1/2)e^(-ix) = y(x).\)
Simplifying even further, we get: y(x) = sin(x)
As a result, given the initial conditions dy(0)/dx = 2 and y(0) = 1, the solution to the above differential equation is y(x) = cos(x).
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Related Questions
Explanation of how we can make (a) subject
Answers
Answer:
Step-by-step explanation:
Please hurry!!!! 50 points!
The diameter of a regulation soccer ball is about 8 3/5 inches. This number was graphed on a number line. (Ill show the number line below) With point could be the point representing the graph of the diameter of the ball?
Answers
Answer:
D
Step-by-step explanation:
8 3/5 is a number greater than 8 but minor than 9
Answer:-9.9 -7 7.9 9.7
Step-by-step explanation:
What are the next 3 numbers in the Geometric sequence 9, 3, 1, 1/3?
Answers
Answer:
The common ratio of this sequence is 1/3, so to find the next three terms we can continue multiplying by 1/3:
1/3 * 1/3 = 1/9
1/9 * 1/3 = 1/27
1/27 * 1/3 = 1/81
Therefore, the next three terms in the sequence are 1/9, 1/27, and 1/81.
1/9, 1/27, and 1/81 are the next 3 numbers in the Geometric sequence 9, 3, 1, 1/3
The given geometric sequence is 9, 3, 1, 1/3.
To find the common ratio, we divide any term by the previous term. Let's use the second and first terms:
3 ÷ 9 = 1/3
The common ratio is 1/3.
To find the next three terms, we can continue multiplying each term by the common ratio.
1/3 * (1/3) = 1/9
(1/3) * (1/9) = 1/27
(1/9) * (1/27) = 1/81
Therefore, the next three terms in the geometric sequence are 1/9, 1/27, and 1/81.
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The Rayhills are buying new windows for their house. The company installs the new windows for $2,800. The total cost for buying the windows and having them installed is $6,484.00. If the Rayhills pay $230 25 per
window, how many windows did they buy?
How many windows did they buy?
windows _
PLEASE HELP PLEASE
Answers
Answer:16
Step-by-step explanation:6484.00-2800 for installation ends up equaling $3684.00
The you divide that answer by $230.25
That equals 16:The answer is 16 windows
Find the slope of the line.
Answers
Answer:
Slope is 3
Step-by-step explanation:
Starting from the point (0,0) move up 3 and over 1. That gives you the slope of 3 over 1 or 3. Slope is rise over run.
What is the value of
\(2 {x}^{ - 2} {y}^{ - 2} \)
for x = 3 and y = -2?
A.
\( \frac{1}{18} \)
B. 72
C.
\(2 {( - 6)}^{ - 4} \)
D. 1/648
Answers
consider+the+following+method.+public+static+void+printsome(int+num1,+int+num2)+{+for+(int+i+=+0;+i+<+num1;+i++)+{+if+(i+%+num2+==+0+
Answers
The given method is a public static void method named "printsome". It takes two integer parameters named "num1" and "num2". The method contains a for loop that initializes an integer variable "i" to 0 and iterates as long as "i" is less than "num1". Within the for loop, there is an if statement that checks if the remainder of "i" divided by "num2" is equal to 0. If the condition is true, the code inside the if statement will be executed.
The given method is:
public static void printSome(int num1, int num2) {
for (int i = 0; i < num1; i++) {
if (i % num2 == 0) {
// code here
}
}
}
In this method:
1. "public" means the method can be accessed from any class.
2. "static" indicates that the method belongs to the class itself and not to any specific instance (object) of the class.
3. "void" denotes that the method does not return any value.
The method takes two integer arguments, num1 and num2. It runs a loop from 0 to num1 (excluding num1). Within the loop, it checks if the current value of 'i' is divisible by num2 (i % num2 == 0). If it is, the code inside the if block will be executed.
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Work out
3/4 times 1/2
Answers
Answer:
\(\frac{3}{4} * \frac{1}{2} = \frac{3}{8}\) alternatively, the answer is 0.375
v=<8,3,5>
w=<2,2,2>
Find the cosine of the angle between v and w
Answers
If v=<8,3,5> w=<2,2,2>, the cosine of the angle between vectors v and w is approximately 0.93294.
To find the cosine of the angle between vectors v and w, we will use the formula:
cos(θ) = (v • w) / (||v|| ||w||)
where θ is the angle between the vectors, v • w is the dot product of v and w, and ||v|| and ||w|| are the magnitudes of v and w, respectively.
First, let's find the dot product (v • w):
v • w = (8 * 2) + (3 * 2) + (5 * 2) = 16 + 6 + 10 = 32
Next, let's find the magnitudes of v and w:
||v|| = √(\(8^2 + 3^2 + 5^2\)) = √(64 + 9 + 25) = √98
||w|| = √(\(2^2 + 2^2 + 2^2\)) = √(4 + 4 + 4) = √12
Now, let's find the cosine of the angle between v and w:
cos(θ) = (32) / (√98 * √12) = 32 / (9.899494936611665 * 3.4641016151377544) ≈ 0.93294
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The cosine of the angle between v and w is 0.9332
To find the cosine of the angle between v and w, we can use the formula:
cos(Ф) = (v.w) / (||v|| ||w||)
where Ф is the angle between v and w, the . represents the dot product, and || || represents the magnitude or length. First, let's calculate the dot product of v and w:
v.w = 8*2 + 3*2 + 5*2 = 32
Next, let's calculate the magnitudes of v and w:
||v|| = sqrt(8^2 + 3^2 + 5^2) = sqrt(98)
||w|| = sqrt(2^2 + 2^2 + 2^2) = sqrt(12)
Now we can substitute these values into the formula:
cos(Ф) = (v.w) / (||v|| ||w||)
cos(Ф) = 32 / (sqrt(98) * sqrt(12))
cos(Ф)= 32 / (sqrt(1176))
cos(Ф) = 32 / 34.29
cos(Ф)= 0.9332
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Determine the value of x.
Answers
Answer:
x = 4.59
Step-by-step explanation:
Since this is a right triangle, we can use trig functions
tan theta = opp / adj
tan 74 = 16/ x
x tan 74 = 16
x = 16/ tan 74
x=4.58792
\(\huge\text{Hey there!}\)
\(\boxed{\boxed{\huge\text{Formula: \boxed{\text{tan } \theta = \mathsf{\dfrac{Opposite}{Adjacent}}}}}}\)
\(\boxed{\huge\text{\boxed{\mathsf{\bold{tan\ 74} = \dfrac{16}{x}}}}}\)
\(\boxed{\boxed{\huge\textsf{x tan 74 equals \bf 16}}}\)
\(\boxed{\huge\text{x = }\boxed{\mathsf{\dfrac{16}{tan \ 74}}}}\)
\(\boxed{\boxed{\huge\text{x = \bf x=4.58792}}}\boxed{\boxed{\huge\text{\bf (ORIGINALLY)}}}\)
\(\boxed{\boxed{\huge\text{x = \boxed{\bf 4.58792\ or \ 4.58 \ is \approx \underline{\underline{4.59}}}}}}\huge\checkmark\)
\(\boxed{\boxed{\huge\text{Therefore, your answer is: \boxed{\textsf{Option C. \bf 4.59}}}}}\huge\checkmark\)
\(\huge\text{\huge\text{Good luck on your assignment \& enjoy your day!}}\)
~\(\boxed{\huge\text{\boxed{\frak{Amphitrite1040:)}}}}\)
20 PTS + Brainliest. Please find the size of the blue area, explain your work.
Answers
here is answer, hope u understand:)
find the area a of the triangle whose sides have the given lengths. (round your answer to three decimal places.) a = 7, b = 5, c = 5
Answers
The area of the triangle with sides a = 7, b = 5, and c = 5 is approximately 12.015 square units, rounded to three decimal places.
To find the area of a triangle with given side lengths a = 7, b = 5, and c = 5, we can use Heron's formula. Heron's formula states that the area (A) of a triangle can be calculated using the semi-perimeter (s) and the side lengths:
1. Calculate the semi-perimeter: s = (a + b + c) / 2
s = (7 + 5 + 5) / 2
s = 17 / 2
s = 8.5
2. Apply Heron's formula: A = √(s * (s - a) * (s - b) * (s - c))
A = √(8.5 * (8.5 - 7) * (8.5 - 5) * (8.5 - 5))
A = √(8.5 * 1.5 * 3.5 * 3.5)
3. Calculate the area:
A ≈ √(144.375)
A ≈ 12.015
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Pls help or I’m gonna fail
Answers
Answer:
SAMEEE
Step-by-step explanation:
Answer:
The equation for a cylinder is V = π r 2 h
Step-by-step explanation:
So this is what your equation would look like:
V = (π)(4)(2)(8)
Pi=3.14
Five out of seven people prefer cranba cranberry juice.How many would prefer cranba cranberry juice in a group of 350 people?
Answers
The people crania cranberry juice in a group of 350 people will be 250.
What is an expression?Expression in maths is defined as the relation of numbers variables and functions by using mathematical signs like addition, subtraction, multiplication and division.
Given that out of seven people, five people prefer crania cranberry juice.
Out of 350 people, the number of people who prefer crania cranberry juice will be calculated as:-
Number of people = ( 5 / 7 ) x 350
Number of people = ( 5 x 50 )
Number of people = 250
Therefore, the people crania cranberry juice will be 250 in a group of 350.
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Round each mixed number to the nearest whole number. Then, estimate the product. 8 9\10 x 1 4\5
Answers
Find anequation for the perpendicular bisector of the line segment whose endpoints are(-4,-5) and (8, -9).
Answers
The equation for the perpendicular bisector will pass through the midpoint of the segment and will have the slope
plz help quick will give brainly
What is the difference between the opposite of 12 and the absolute value of 12? Use a number line to explain.
Answers
Answer:
Opposite 12= -12
Absolute value of 12: Just -12.
Step-by-step explanation:
I don't know.
A submarine at 21 feet below sea level suddenly moves up about 6 feet. At what depth is
the submarine located now?
Answers
Answer:
The submarine is 15 feet below sea level.
Step-by-step explanation:
Find the volume of the solid lying under the circular paraboloid z = x2 + y2 and above the rectangle R = (-4,4] x [-6,6). 1. 2496 2. 1664 3. 1248 4. 960 5. 640
Answers
According to the question we have the correct answer is option 2, with a volume of 1664 cubic units.
The volume of the solid lying under the circular paraboloid z = x^2 + y^2 and above the rectangle R = (-4, 4] x [-6, 6] can be found using a double integral. First, set up the integral with respect to x and y over the given rectangular region:
Volume = ∬(x^2 + y^2) dA
To evaluate this integral, we will use the limits of integration for x from -4 to 4, and for y from -6 to 6:
Volume = ∫(from -4 to 4) ∫(from -6 to 6) (x^2 + y^2) dy dx
Now, integrate with respect to y:
Volume = ∫(from -4 to 4) [(y^3)/3 + y*(x^2)](from -6 to 6) dx
Evaluate the integral at the limits of integration for y:
Volume = ∫(from -4 to 4) [72 + 12x^2] dx
Next, integrate with respect to x:
Volume = [(4x^3)/3 + 4x*(72)](from -4 to 4)
Evaluate the integral at the limits of integration for x:
Volume = 1664
Therefore, the correct answer is option 2, with a volume of 1664 cubic units.
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The figure that will be formed if two 45° 45° 90° setsquares are put together is _________.
Answers
Organizational culture is set by a. the manager b. the ethics committee c. the engineer d. none of the given options
Answers
Organizational culture is defined as the common beliefs, values, attitudes, customs, behaviors, and traditions that characterize a specific organization and determine the manner in which it functions. Organizational culture is set by the manager.
In a corporate or business environment, organizational culture can influence the daily operations of employees. It is the responsibility of managers to create a positive culture that emphasizes teamwork, respect, integrity, and accountability. The manager is an essential individual responsible for establishing and maintaining the organization's culture, which will ultimately define the employee's attitudes, behaviors, and productivity levels. He or she sets the tone for the workplace by creating an environment that fosters collaboration, innovation, and success. Employees need to feel connected to their workplace and colleagues to be motivated to do their best work. If a manager promotes a culture of fear, competition, or dishonesty, employees may become unmotivated or unproductive. An effective manager understands the importance of creating a positive workplace culture and works hard to establish and maintain it. Managers can establish a positive culture by encouraging open communication, providing regular feedback and recognition, fostering a sense of teamwork, creating opportunities for professional development, and setting high standards for performance. Managers must lead by example and demonstrate the behaviors and attitudes that they expect from their employees. They must hold themselves and others accountable for their actions, communicate expectations clearly, and provide support when needed. A positive organizational culture will enable an organization to attract and retain top talent, increase employee engagement, and promote collaboration and innovation.
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Can someone please help me with this?
Find the x- and y-intercepts of the equation.
4x − 2y = 2
(A) (1/2,0). (0,1)
(B). (2, 0), (0, −1)
(C). (2, 0), (0, −1)
(D). (1/2, 0), (0, −1)
Answers
Answer:
Option D
Step-by-step explanation:
Convert to slope-intercept form:
\(4x-2y=2\)
\(-2y=2-4x\)
\(y=-1+2x\)
\(y=2x-1\)
Find the x-intercept by plugging in y=0:
\(y=2x-1\)
\(0=2x-1\)
\(1=2x\)
\(\frac{1}{2}=x\)
\(x=\frac{1}{2}\)
Therefore, the x-intercept is \((\frac{1}{2},0)\)
Find the y-intercept by plugging in x=0:
\(y=2x-1\)
\(y=2(0)-1\)
\(y=0-1\)
\(y=-1\)
Therefore, the y-intercept is \((0,-1)\)
So, the correct answer is option D.
Answer:
D
Step-by-step explanation:
1. To find the x intercepts, set y equal to 0 and solve for x
4x - 2y = 2
4x -2(0) = 2
4x - 0 = 2
4x = 2
x = 1/2
2. To find the y intercepts, set x equal to 0 and solve for y
4x - 2y = 2
4(0) - 2y = 2
0 - 2y = 2
-2y = 2
y = -1
....
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Answers
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Answer:
Step-by-step explanation:
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a sample consists of the following data: 7, 11, 12, 18, 20, 22, 43. Using the three standard deviation criterion, the last observation (x=43) would be considered an outlier
a. true
b. false
Answers
The statement "Using the three standard deviation criterion, the last observation (x=43) would be considered an outlier" is true.
Given data:
7, 11, 12, 18, 20, 22, 43.
To find out whether the last observation is an outlier or not, let's use the three standard deviation criterion.
That is, if a data value is more than three standard deviations from the mean, then it is considered an outlier.
The formula to find standard deviation is:
S.D = \sqrt{\frac{\sum_{i=1}^{N}(x_i-\bar{x})^2}{N-1}}
Where, N = sample size,
x = each value of the data set,
\bar{x} = mean of the data set
To find the mean of the given data set, add all the numbers and divide the sum by the number of terms:
Mean = $\frac{7+11+12+18+20+22+43}{7}$
= $\frac{133}{7}$
= 19
Now, calculate the standard deviation:
$(7-19)^2 + (11-19)^2 + (12-19)^2 + (18-19)^2 + (20-19)^2 + (22-19)^2 + (43-19)^2$= 1442S.D
= $\sqrt{\frac{1442}{7-1}}$
≈ 10.31
To determine whether the value of x = 43 is an outlier, we need to compare it with the mean and the standard deviation.
Therefore, compute the z-score for the last observation (x=43).Z-score = $\frac{x-\bar{x}}{S.D}$
= $\frac{43-19}{10.31}$
= 2.32
Since the absolute value of z-score > 3, the value of x = 43 is considered an outlier.
Therefore, the statement "Using the three standard deviation criterion, the last observation (x=43) would be considered an outlier" is true.
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Pls help me I don’t which on it is
Answers
The table of values below represents a linear function and shows the height of a tree since it was transplanted. What was the height of the tree when it was transplanted? Height of a Tree Since It Was Transplanted Years Since the Tree Was Transplanted 4 4.5 5 5.5 6 Height (feet) 12 13 14 15 16 1 foot 2 feet 4 feet 8 feet
Answers
Answer:
I am pretty sure that it is 4.
Step-by-step explanation: Got it right on edge2020
Answer:
4
Step-by-step explanation:
i took on edge
if a wave has a length of 30 feet, what is the depth? question 2 options: a) 20 feet b) 15 feet c) 10 feet d) 5 feet
Answers
The length of a wave does not directly determine its depth. The depth of a wave is influenced by various factors, including the shape and size of the seafloor,
The temperature and salinity of the water, and the strength and direction of the current. Therefore, without additional information about these factors, it is impossible to accurately determine the depth of a wave based solely on its length. I suggest seeking the advice of an expert in oceanography or physics for a more comprehensive explanation.
The length of a wave is the distance between two successive points in the same phase, such as from crest to crest or trough to trough. In your case, the wave length is given as 30 feet. However, without additional information about the wave, we cannot determine its depth solely based on its length.
Depth refers to the vertical distance from the surface to the lowest point of a wave (trough), and it's not directly related to the wave length. Therefore, we cannot accurately choose an option (a, b, c, or d) for the depth based on the provided information.
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Write an inequality to represent the following sentence:
“Six is at least four more than a number.”
Answers
Answer:6 < 4+n
Step-by-step explanation:at least symbol is < . and more than means add .
On a coordinate graph the following points of a rectangle are given. A(0,2) B (0,-2) C (3,-2) 4 Where do the coordinates of Dneed to be to form the rectangle? 0 (3,2) O (2, 3) 0 (1,3) 0 (3,1)
Answers
Answer:
(3, 2)
Step-by-step explanation:
Please mark me brainliest if this helped! :P
what are the quotient and remainder when −13 is divided by 6? (15 pts).
Answers
The quotient and remainder of -13 by 6 is -2 and -1 respectively.
To find the quotient and remainder when -13 is divided by 6, we can use the following steps:
Divide -13 by 6: -13 ÷ 6 = -2.1666666666666665Round the quotient down to the nearest whole number: -2.1666666666666665 ≈ -2Multiply the quotient by the divisor: -2 × 6 = -12Subtract the product from the dividend: -13 - (-12) = -1Therefore, the quotient when -13 is divided by 6 is -2 and the remainder is -1.
What is a quotient?In arithmetic, when we talk about a quotient, we refer to the result of a division between two numbers. This is mostly known to be the integer part of a fraction or, as mentioned above, the result of a division.
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