Use The Laplace Transform To Solve The Following Initial Value Problem:Y" -8y'+20y=0 Y(0)=1,Y'(0)=4First, Using Y For The Laplace Transform Of Y(T) , I.e., L{Y(T)} ,Find The Equation You Get By Taking The Laplace Transform Of The Differential Equation: _________________=0Now Solve For Y(S)=?By Completing The Square In The Denominator And Inverting The
Use the Laplace transform to solve the following initial value problem:
y" -8y'+20y=0 y(0)=1,y'(0)=4
First, using Y for the Laplace transform of Y(t) , i.e., L{Y(t)} ,
find the equation you get by taking the Laplace transform of the differential equation: _________________=0
Answers
Taking the Laplace transform of the differential equation y" -8y'+20y=0, we get:
s^2Y(s) -sy(0) -y'(0) -8(sY(s) -y(0)) +20Y(s) = 0
Substituting y(0)=1 and y'(0)=4, we get:
s^2Y(s) -s -4 -8sY(s) +8 +20Y(s) = 0
Simplifying, we get:
(s^2-8s+20)Y(s) = s+12
Therefore, we have:
Y(s) = (s+12)/(s^2-8s+20)
Next, we need to find the inverse Laplace transform of Y(s) to obtain the solution y(t). However, we can simplify the expression for Y(s) using partial fractions:
Y(s) = (s+12)/[(s-4)^2 + 4]
Using partial fractions, we can write:
Y(s) = A/(s-4) + [B(s-4)+C]/[(s-4)^2 + 4]
Multiplying both sides by the denominator, we get:
(s+12) = A[(s-4)^2 + 4] + B(s-4)+C
Setting s=4, we get:
16A = 16B
Therefore, A=B.
Differentiating both sides with respect to s, we get:
1 = 2A(s-4) + B(s-4) + (B(s-4)+C)
Setting s=4, we get:
1 = 2B
Therefore, B=1/2 and A=1/2.
Substituting these values into the partial fractions expression, we get:
Y(s) = 1/2/(s-4) + [1/2(s-4)+C]/[(s-4)^2 + 4]
To find C, we can multiply both sides by [(s-4)^2 + 4] and then take the limit as s approaches infinity, which gives:
C = 0
Therefore, we have:
Y(s) = 1/2/(s-4) + 1/2(s-4)/[(s-4)^2 + 4]
Taking the inverse Laplace transform, we get:
y(t) = (1/2)e^(4t)cos(2t) + (1/4)e^(4t)sin(2t)
Therefore, the solution to the initial value problem is:
y(t) = (1/2)e^(4t)cos(2t) + (1/4)e^(4t)sin(2t)
Note that the domain of y(t) is t >= 0.
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Related Questions
What is the equation of the line through 1 2 which makes equal intercepts on the axis?
Answers
The equation of the line through \((1,2)\) which make equal intercepts on the axis \(x+y=3\)
The equation of the line through (1,2) makes an equal intercept on the axis
The formula of the intercept form is
\(\frac{x}{a} +\frac{y}{b} =1\)
If they make an equal intercept
\(a=b\\\frac{x}{a} +\frac{y}{a} =1\\x+y=a\)
Put the value of the point in the axis, and we get.
\(1+2=a\\a=3\)
Put the value in the equation, and we get.
\(x+y=3\)
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Written as a simplified polynomial in standard form, what is the result when (2x-4)^2 is subtracted from 4?
Answers
Answer:
?
Step-by-step explanation:
???
Samir paints 13 of a fence in 23 hour. He continues to paint the fence at the same rate. Which expression can be used to determine the fraction of the fence that Samir will paint in 1 hour?
Answers
Answer:
Step-by-step explanation:Samir should just paint better and quicker
Samir paints 1/3 of a fence in 2/3 hours. To find his rate per hour, you would divide the fraction of the fence painted by the time taken. Thus, the desired expression is (1/3) ÷ (2/3) = 1/2, meaning Samir paints half of the fence in one hour.
Explanation:The question is about determining the rate at which Samir paints a fence. According to the question, Samir paints 1/3 or 13 of a fence in 2/3 or 23 hours. To find out how much fence Samir paints in 1 hour, or his rate, we divide the fraction of the fence painted by the time taken.
So the expression would be (1/3) ÷ (2/3). In essence, we're dividing the fraction of the fence painted by the corresponding time. When you divide fractions, you multiply the first fraction by the reciprocal of the second fraction. Thus, it becomes (1/3) * (3/2) = 1/2. So, Samir would be able to paint 1/2 of the fence in one hour if he continues to paint at the same rate.
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Starting at the same location, Jack is running Northwards at 2ft/s while Kim is running Westwards at 3ft/s. How fast are they moving apart from each other at 10 seconds? A. 5
8
ft
B. 13
s
ft
C.
5
s
ft
D.
13
s
ft
E. NO correct choices
Answers
The Answer is E.NO correct choices
The rate at which Jack and Kim are moving apart from each other after 10 seconds is 0 ft/s.
To determine how fast Jack and Kim are moving apart from each other, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, Jack's motion in the North direction and Kim's motion in the West direction form a right triangle. The sides of the triangle represent the distances each person has traveled after 10 seconds.
Jack's distance = speed * time = 2 ft/s * 10 s = 20 ft
Kim's distance = speed * time = 3 ft/s * 10 s = 30 ft
Now, let's calculate the distance between them using the Pythagorean theorem:
\(Distance^2 = Jack's distance^2 + Kim's distance^2\)
\(Distance^2 = 20^2 + 30^2\)
\(Distance^2 = 400 + 900\)
\(Distance^2 = 1300\)
\(Distance = \sqrt{1300}\)
Distance ≈ 36.06 ft
Therefore, the distance between Jack and Kim after 10 seconds is approximately 36.06 ft.
Since the question asks for the rate at which they are moving apart, we need to calculate the rate of change of this distance over time.
Rate = Change in distance / Change in time
To find the change in distance, we can differentiate the equation Distance = sqrt(1300) with respect to time:
d(Distance)/dt = (1/2) *\((1300)^(-1/2)\) * d(1300)/dt
d(Distance)/dt = (1/2) * \((1300)^(-1/2)\) * 0
d(Distance)/dt = 0
Therefore, the rate at which Jack and Kim are moving apart from each other after 10 seconds is 0 ft/s.
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20)
A single card is chosen at random from a standard deck of 52 playing cards. Which BEST describes the probability of drawing a king
from the deck?
Answers
The best description of the probability of drawing a king from the deck is 1 out of 13, or 1/13.
The probability of drawing a king from a standard deck of 52 playing cards can be determined by dividing the number of favorable outcomes (number of kings) by the total number of possible outcomes (total number of cards in the deck).
In a standard deck, there are 4 kings (one king for each suit: hearts, diamonds, clubs, and spades). Therefore, the number of favorable outcomes is 4.
The total number of possible outcomes is 52 (the total number of cards in the deck).
So, the probability of drawing a king is:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 4 / 52
Simplifying the fraction gives us:
Probability = 1 / 13
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help meeeeeeeeeeeee pleaseeee rnnn!!!
Answers
The Vamers live on a corner lot. Often, children cut across their lot to save walking distance. the walking distance that is saved by cutting across their property instead of walking around the lot is 20 feet (to the nearest integer)
How to determine the working distance savedinformation given in the question
hypotenuse = 48 feet
opposite = x
adjacent = x + 6
The problem is solved using the Pythagoras theorem is applicable to right angle triangle. the formula of the theorem is
hypotenuse² = opposite² + adjacent²
plugging the values as in the problem
let x be the required distance
48² = x² + (x+6)²
2304 = x² + x² + 12x + 36
equating to zero
0 = x² + x² + 12x + 36 - 2304
0 = 2x² + 12x - 2268
2x² + 12x - 2268 = 0
x = 30.808 OR -36.808
taking the positive value, x = 30.808 feet
adjacent = x + 6 = 30.808 + 6 = 36.808
The total distance
= adjacent + opposite
= 20.808 + 36.808
= 67.616
walking distance saved
= total distance - hypotenuse
= 67.616 - 48
= 19.616
= 19.62 feet (to the nearest hundredth)
= 20 feet (to the nearest integer)
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Please help!!! Angles!
Answers
Answer:
m∠JKM = 63°
m∠MKL = 27°
Step-by-step explanation:
Since ∠JKL is a right angle. This means that by summing up both m∠JKM and m∠MKL will result in the same as ∠JKL figure. Thus, m∠JKM + m∠MKL = m∠JKL which is 90° by a right angle definition.
\(\displaystyle{\left(12x+3\right)+\left(6x-3\right) = 90}\)
Solve the equation for x:
\(\displaystyle{12x+3+6x-3 = 90}\\\\\displaystyle{18x=90}\\\\\displaystyle{x=5}\)
We know that x = 5. Next, we are going to substitute x = 5 in m∠JKM and m∠MKL. Thus,
m∠JKM = 12(5) + 3 = 60 - 3 = 63°
m∠MKL = 6(5) - 3 = 30 - 3 = 27°
*PLEASE ANSWER!!!, I DONT GET IT!! HELP!!*
In the United States, the official poverty rate is the percentage of residents who _____.
A.) are unable to purchase food or water from local sources
B.) are earning an individual income that is less than the threshold set by government
C.) are earning a household income that is less than the threshold set by government
D.) are unable to purchase food or water anywhere in the country
Answers
Answer:
1 are earning a household income that is less than the threshold set by government
2 The Supplemental Poverty Measure includes nonmonetary benefits like housing subsidies and tax credits, while the official poverty measure does not.
3 distribution of total GDP among the working population
4 Social Security income
5 progressive tax
OCA Poverty and Resource Distribution Quick Check
The official poverty rate is the percentage of residents who are earning a household income that is less than the threshold set by the government. So, option C is correct.
What is the poverty threshold?The poverty threshold is the minimum level of income of a person in a country. It may be adjusted annually.
Poverty can be calculated by the total cost of all the basic needs or resources that an average human can consume in a year.
Poverty depends on various factors such as lack of education, economic crisis, climate changes, lack of resources, etc.
The Supplemental Poverty Measure includes nonmonetary benefits like housing subsidies and tax credits, while the official poverty measure does not.
Distribution of total GDP among the working population
The official poverty rate is the percentage of residents who are earning a household income that is less than the threshold set by the government.
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Research Study: We want to figure out what characteristics of boys are most attractive to high-school girls. High-school girls were randomly assigned three groups to hear about a guy with trendy clothes, a nice car, or a confident attitude. There were 9 girls in each group. They gave ratings on a scale from 1 to 10, with 1 being "not interested" and 10 being "very interested."
Question: Is there a significant difference between the attractiveness ratings given by the three groups of girls?
What is the factor?
What are the levels of the factor?
Step 1: State Hypotheses
Null:
Research:
Step 2: Determine Comparison Distribution
What is the distribution shape and degrees of freedom for the comparison distribution?
Step 3: Set the Criteria for a Decision
Alpha = .05
dfB =
dfW=
Critical value:
Is the rejection region placed in the upper or lower tail? How do you know?
Step 3: Compute the Test Statistic
Trendy Clothes
Nice Car
Confident Attitude
8
6
10
6
7
9
9
8
10
6
6
10
8
4
8
6
5
9
7
5
10
6
6
9
6
8
8
Step 5: Make a Decision
Reject/Fail to Reject the null?
What does this mean (be sure to talk about significance)?
Answers
Test statistic Z= 0.13008 < 1.96 at 0.10 level of significance.
The null hypothesis is accepted.
There is no significant difference between the attractiveness ratings given by the three groups of girls.
Surveyed two random samples of 390 men and 360 women who were tested.
first sample percentage,
p1 = 360 ÷ 390 = 0.9230
second sample percentage,
p2 = 47 ÷ 52 = 0.9030
There is no change, according to the null hypothesis the percentage of males who have positive tests is significantly from the percentage of women who have positive tests.
Different Hypotheses:
There is a discrepancy in the percentage of schoolgirls who heard about a male with stylish clothes, a great automobile, or a confident demeanor across the three groups that were randomly allocated.
Z = (p1 - p2) ÷ √(PQ((1 ÷ n1) + (1 ÷ n2)))
where
p = (n1p1 + n2p2) ÷ n1 + n2
p = 0.920
Z = (0.9230 - 0.9030) ÷ √(0.9230((1 ÷ 390) + (1 ÷ 52)))
Test statistic Z = 0.13008
Level of significance = 0.10
The critical value Z₀.₁₀ = 1.645
Test statistic Z=0.13008 < 1.645 at 0.1 level of significance
The null hypothesis is accepted.
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wich is bigger 51.243 or 5.1243
Answers
Answer:
The number 51.243 is bigger than 5.1243.
Step-by-step explanation:
To compare two decimal numbers, we start by looking at the digits to the left of the decimal point. In this case, both numbers have the same digit in the tens place, which is 5. So we need to look at the next digit to the right of the decimal point.In 51.243, the digit to the right of the decimal point is 2, which is larger than the digit to the right of the decimal point in 5.1243, which is 1. Therefore, we can conclude that 51.243 is bigger than 5.1243.
Answer:
51.243
Step-by-step explanation:
because 51 is greater than 5.1
Please help!! 55 points!!
Answers
B. 7/20 is the correct answer
Please help! Will mark Branliest!
In Game 2, your score started at 2 and doubled every time. Two expressions below represent your score after 12 hits. Select both.
Answers
Answer:
B and D
Why are both of the answers to the last question correct?
An exponent is a short way to write repeated multiplication. So 212 means the same as 2 • 2 • . . . • 2 • 2.
Directions: Evaluate the following logarithmic functions.
1. log100.1
2. log5125
3. log864
4. log1/3 9
5. log1/21/2
6. log497
7. log9 3
8. log3 81
9. log4 8
10. log100 10,000
Answers
The result of the logarithmic functions by logarithm tables are listed below:
- 232- 211 / 21 / 243 / 22How to evaluate logarithmic functions
This question can be resolved by means of logarithm properties and logarithm tables or numerical methods. Now we proceed to evaluate each logarithmic functions:
Case 1:
㏒₁₀ 0.1
By using rational numbers:
㏒₁₀ (1 / 100)
By logarithm properties:
㏒₁₀ 1 - ㏒₁₀ 100
By logarithm tables:
0 - 2
- 2
Case 2:
㏒₅ 125
By logarithm tables:
3
Case 3:
㏒₈ 64
By logarithm tables:
2
Case 4:
\(\log_{\frac{1}{3} } 9\)
By logarithm tables:
- 2
Case 5:
\(\log_{\frac{1}{2}} \frac{1}{2}\)
By logarithm tables:
1
Case 6:
㏒₄₉ 7
By logarithm tables:
1 / 2
Case 7:
㏒₉ 3
By logarithm tables:
1 / 2
Case 8:
㏒₃ 81
By logarithm tables:
4
Case 9:
㏒₄ 8
By logarithm tables:
3 / 2
Case 10:
㏒₁₀₀ 10,000
By logarithm tables:
2
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The average salary of an accountant is $ 71,000 a year. He just finished and his training which will increase his salary by 20%. How much more money he will make in next 10 years as compared to what he was earning without the training?
Answers
Answer:
After 10 years he will make 142 000$ more compared to what was earning without training
Please help me!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
B
Step-by-step explanation:
its hard to explain, but just trust me bro. lol.
Please help explanation if possible
Answers
Explanation:
y < mx + b
b is y-intercept
mx is slope
To find b I count from (0,0) upward a long the y-axis where the line intercept at point (0,5). So 5 - 0 = 5 therefore, b = 5
Simplify the expression by combining like terms.
Write the terms in alphabetical order of the
variables.
5x - 2y - 6z - 10x - 14y - 18z
Answers
Answer:
-5x - 16y - 24z
Step-by-step explanation:
5x - 2y - 6z - 10x - 14y - 18z
5x - 10x - 2y - 14y - 6z - 18z
-5x - 16y - 24z
-16 + 4x^2 = 0
Please help. If possible can I get a step by step ?
Answers
Step-by-step explanation:
\( - 16 + 4 {x}^{2} = 0 \\ 4 {x}^{2} = 16 \\ {x}^{2} = \frac{16}{4} \\ {x}^{2} = 4 \\ x = \sqrt{4 } \\ x = 2\)
CAN ANYONE PLZZ TELL ME THE SQUARE OF ALL THE NUMBERS IN THE CLOCK ?
PLZZ
Answers
Answer:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Step-by-step explanation:
Represent the following sentence as an algebraic expression, where "a number" is the letter x.
\text{A number is subtracted from 3.}
A number is subtracted from 3.
Answers
The required algebraic expression is x-3
What is an Algebraic Expression?An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms.
Given here: The statement "A number subtracted from 3 "
Thus translating the given text we get the following:
Let the number be x
And we have have to subtract 3 from x
Thus the algebraic expression is given by x-3
Hence, The required algebraic expression is x-3
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What is the equation of the line that passes through the point (-3,2) and has a slope of 2/3
Answers
Answer: y = 2/3 -2
Step-by-step explanation: using a graph you can plot the point (-3, 2).
After this, you can use the slope of 2/3 to graph the starting point. You go 3 to the left and 2 up going through the y axis of -2. using y = mx + b, you can plug in the slope of 2/3 and the starting point of -2.
Express 3.23×10 ^6
in standard decimal notation.
Answers
The number 3.23×10^6, expressed in standard decimal notation, is 3,230,000. In scientific notation, a number is expressed as the product of a coefficient and a power of 10.
In this case, the coefficient is 3.23, and the power of 10 is 6. To convert this number into standard decimal notation, we need to multiply the coefficient by the corresponding power of 10. The power of 10, in this case, is 10^6, which means 10 raised to the power of 6. This is equivalent to 1 followed by six zeros: 1,000,000. To obtain the standard decimal notation, we multiply the coefficient, 3.23, by 1,000,000.
Calculating the multiplication, we have:
3.23 × 1,000,000 = 3,230,000.
Hence, the number 3.23×10^6, expressed in standard decimal notation, is 3,230,000. This means that the number is equal to 3.23 million.
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The function g is a transformation of f. If g has a y-intercept at -1, which of the following functions could represent g?
A.
g(x) = f(x) + 4
B.
g(x) = f(x + 4)
C.
g(x) = f(x) - 1
D.
g(x) = f(x - 1)
Answers
Simplify: - \(6\sqrt{63} -4\sqrt{28}\)
Answers
Answer:10\(\sqrt{7}\)
Step-by-step explanation:
A library has 500 books. There were history, maths and science books. The ratio of the history to maths
to science books is 2:3:5. If 20 history, 40 maths, and 10 science books were added in the library,
find the new ratio of the history to the maths to the science books.
Pls help
Answers
Answer:
The new ratio of the history to the maths to the science books would be 22:43:55. With the addition of 20 history, 40 maths, and 10 science books, the total number of books in the library is now 570. The new ratio of the history to the maths to the science books is therefore 22:43:55.
Step-by-step explanation:
Need help ASAP! Please
Answers
Answer:
112
Step-by-step explanation:
20 x 20 is 40
6 x 9 is 54
6 x 3 is 18
54 + 18 is 72
72 + 40 is 112
Answer:
472
Step-by-step explanation:
Use the order of operations PEMDAS
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
9 + 3 = 12
20^2 = 400
6 * 12 = 72
400 + 72 = 472
hope this helped! :)
The distance around a meter crater is 9755 ft. Find the diameter of the crater. Use 22/7
Answers
Answer:
\(\frac{68285}{44}\) or \(1551.93181818...\)
Step-by-step explanation:
\(\frac{9755}{2(\frac{22}{7})}\\\\\)
How to find the variance of uniform distribution?
Answers
We can find the variance of a uniform distribution, using the following formula: Variance = (b - a)² / 12. Here: a is the lower limit of the uniform distribution, b is the upper limit of the uniform distribution.
The formula for the variance of a uniform distribution is based on the fact that the variance of a continuous uniform distribution is equal to the square of the range (i.e., the difference between the upper and lower limits) divided by 12.
For example, let's say we have a uniform distribution between 0 and 10. To find the variance, we would use the formula:
Variance = (10 - 0)² / 12
Variance = 100 / 12
Variance = 8.33
Therefore, the variance of a uniform distribution between 0 and 10 is 8.33.
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which number is an integer? 6 4.2 3/4 5 1/2
Answers
Answer:
5, 6
Step-by-step explanation :
PLEASE HELP THIS IS MY LAST QUESTION
If the correlation coefficient for the data shown in the table is -1, A should be what value?
Time (hours) (x)
4 1
3 6 5 7
Distance from destination (miles) (y) 1,000 A 1,060 940 760 820 690
920
880
800
none of these
O 1030
2
Answers
Answer:
Step-by-step explanation:
Here is the answer
The correlation coefficient measures the strength of the linear relationship between two variables, and it ranges from -1 to 1. A correlation coefficient of -1 indicates a perfect negative linear relationship between the variables, which means that as one variable increases, the other decreases at a constant rate.
To find the value of A in this scenario, we need to look for a perfect negative linear relationship between the two variables, time (x) and distance from destination (y). The table shows that as time increases, the distance from the destination decreases, but we need to find the exact rate of change.
We can calculate the rate of change by finding the slope of the line that represents the relationship between time and distance. We can use the formula for the slope of a line, which is:
slope = (change in y) / (change in x)
If we choose the first and last points in the table, we get:
slope = (690 - 1060) / (7 - 1) = -70
This means that for every hour of time that passes, the distance from the destination decreases by 70 miles. Therefore, if the correlation coefficient is -1, we should see a perfect negative linear relationship between time and distance, with a slope of -70.
To check if A is the correct value, we can use the formula for the equation of a line in slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. We can plug in the values of m and b and solve for A:
y = -70x + b
If we use the first point in the table, where x = 4 and y = 1000, we get:
1000 = -70(4) + b
b = 1220
So the equation of the line is:
y = -70x + 1220
If we plug in the values of x for the remaining points in the table, we get:
y = 1030 when x = 0
y = 880 when x = 2
y = 800 when x = 3
y = A when x = 5
y = 760 when x = 6
To find the value of A, we can plug in the corresponding value of y and solve for A:
1030 = -70(0) + 1220
880 = -70(2) + 1220
800 = -70(3) + 1220
760 = -70(6) + 1220
A = -70(5) + 1220 = 850
Therefore, the value of A should be 850 if the correlation coefficient for the data shown in the table is -1.