Use the Laplace transform to solve the given initial-value problem. y'' − 3y' = 4e2t − 2e−t, y(0) = 1, y'(0) = −1 y(t) =
Answers
The solution to the initial-value problem is using Laplace transform
y(t) = (-4/3 + 2/3 \(e^{(3t)\)) u(t) + (1/3 - 2/3 \(e^{(3t)\)) u(t-0)
To solve the initial value problem using Laplace transform, we first apply the transform to both sides of the differential equation.
Taking the Laplace transform of the left-hand side, we get:
L{y'' - 3y'} = L\({d^2y/dt^2}\) - 3L{dy/dt} = \(s^2Y(s)\) - sy(0) - y'(0) - 3(sY(s) - y(0)) = \(s^2\)Y(s) - s - 1 - 3sY(s) + 3
Simplifying this expression, we get:
L{y'' - 3y'} = (\(s^2\) - 3s)Y(s) - s + 2
Next, we take the Laplace transform of the right-hand side of the differential equation. Using the Laplace transform of \(e^{at\), we have:
L{\(4e^{2t\)} = 4/(s-2)
L{\(2e^{-t\)} = 2/(s+1)
So, the Laplace transform of the right-hand side is:
\(L{4e^{2t - 2e^{-t}} = 4/(s-2) - 2/(s+1)\)
Putting it all together, we have:
(\(s^2\) - 3s)Y(s) - s + 2 = 4/(s-2) - 2/(s+1)
Simplifying this expression and solving for Y(s), we get:
Y(s) = [(4/(s-2) - 2/(s+1)) + s - 2]/(\(s^2\) - 3s)
Y(s) = [4/(s-2) - 2/(s+1) + s - 2]/(s(s-3))
Now, we use partial fraction decomposition to simplify this expression. We write:
Y(s) = A/s + B/(s-3)
Multiplying both sides by s(s-3), we get:
Y(s) = A(s-3) + Bs
Substituting this expression into the previous equation, we get:
A(s-3) + Bs = [4/(s-2) - 2/(s+1) + s - 2]/(s(s-3))
Equating the coefficients of the like terms on both sides, we get:
A - 3B = -2
B = 2/3
A = -4/3
So, the Laplace transform of the solution y(t) is:
Y(s) = -4/(3s) + 2/(3(s-3))
Taking the inverse Laplace transform of this expression, we get:
y(t) = (-4/3 + 2/3 \(e^{(3t)\)) u(t)
where u(t) is the unit step function.
Using the initial conditions y(0) = 1 and y'(0) = -1, we can solve for the constants:
y(0) = -4/3 + 2/3 = 1
y'(t) = \(2e^{(3t)}/3\)
y'(0) = 2/3 - 1 = -1/3
Therefore, the solution to the initial-value problem is:
y(t) = (-4/3 + 2/3 \(e^{(3t)\)) u(t) + (1/3 - 2/3 \(e^{(3t)\)) u(t-0)
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Related Questions
find the x and y intercepts of 2x-4y=7
Answers
Answer:
x = (7/2, 0)
y = (0, -7/4)
Step-by-step explanation:
Answer:
x-intercept: (7/2 , 0)
y-intercept: (0, -7/4)
Step-by-step explanation:
To find the x-intercept, plug in 0 for y and solve. To find the y-intercept, plug in o for x and slove.
What is the perimeter of the house is c=5
I’m marking brainliest.
Answers
Answer:
132
Step-by-step explanation:
C=5
3c - 2 = 13
2c = 10
4c + 1 = 21
4c + 2 = 22
length = (3c- 2) + (4c + 1)
length = 13 + 21
length = 34
width = (4c + 2) + (2c)
width = 22 + 10
width = 32
Perimeter
p = 2l + 2w
p = 2*34 + 2*32
p = 68 + 64
p = 132
Holly finds that (11m – 13n 6mn) – (10m – 7n 3mn) = m – 20n 9mn. what error did holly make?
Answers
The error holly made is instead of -6n there is -20n and in place of 3mn there is 9mn.
What is equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Given: (11m – 13n + 6mn) – (10m – 7n + 3mn)
= 11m – 13n + 6mn -10m +7n - 3mn
= 11m- 10m -13n + 7n + 6mn - 3mn
=m - 6n + 3mn
Hence, the correct answer is m - 6n + 3mn
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a fair, six-sided die is rolled repeatedly until either two prime numbers or two even numbers are rolled. what is the expected number of ones that are rolled?
Answers
The probability number of times to roll the number cube to result in a prime number 12 times is 24
How to determine the number of times?
The given parameters are:
Number cube = 6 sides
Number of times= 12
The probability of obtaining a prime number in a six-sided die is:
P(Prime) = 3/6
The number of times (n) is then calculated as:
p * n = 12
Substitute known values
3/6 * n = 12
Evaluate the fraction
1/2 * n = 12
Multiply both sides by 2
n = 24
Hence, the number of times is 24
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If46 grams of ch4 react with 32 grams of o2 according to the following
Answers
To determine the limiting reactant and calculate the amount of product formed when 46 grams of CH4 (methane) react with 32 grams of O2 (oxygen), we need to use the concept of stoichiometry and compare the reactants' ratios in the balanced chemical equation.
To begin, we need to balance the chemical equation for the reaction between CH4 and O2. The balanced equation is:
CH4 + 2O2 -> CO2 + 2H2O
According to the balanced equation, the molar ratio between CH4 and O2 is 1:2. This means that for every 1 mole of CH4, we need 2 moles of O2 to react completely and produce the expected amount of product.
Next, we calculate the number of moles for each reactant. The molar mass of CH4 is 16 g/mol, so:
Moles of CH4 = Mass of CH4 / Molar mass of CH4
= 46 g / 16 g/mol
= 2.875 mol
Similarly, the molar mass of O2 is 32 g/mol, so:
Moles of O2 = Mass of O2 / Molar mass of O2
= 32 g / 32 g/mol
= 1 mol
Now, we compare the moles of CH4 and O2 to their stoichiometric ratios. Since the stoichiometric ratio between CH4 and O2 is 1:2, it means that 2 moles of O2 are required for 1 mole of CH4.
As we have 2.875 moles of CH4 and 1 mole of O2, we can see that we have an excess of CH4 and a limiting amount of O2. Therefore, O2 is the limiting reactant in this reaction.
To calculate the amount of product formed, we use the stoichiometric ratio between CH4 and CO2. According to the balanced equation, 1 mole of CH4 produces 1 mole of CO2.
Hence, the amount of CO2 produced will be equal to the moles of the limiting reactant, which in this case is O2. Therefore, the amount of CO2 produced is 1 mol.
By using the concept of stoichiometry and comparing the moles of reactants to their respective stoichiometric ratios, we determine the limiting reactant (O2) and find that 1 mole of CO2 is formed as a product.
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Given f(x) = x^2 - 1 and g(x) = 5x^2+2, find (f/g)(2)
can someone help me figure out how to do this? i need to show work (the “^2” means it’s squared by the way)
Answers
\(\frac{f}{g}(2) = \frac{2}{11}\\\)
Step-by-step explanation:\(\frac{f}{g}(2) = \frac{f(2)}{g(2)} \\ \frac{f}{g}(2) = \frac{(2)^2}{5(2)^2 +2} \\ \frac{f}{g}(2) = \frac{4}{5(4) +2} \\ \frac{f}{g}(2) = \frac{4}{20 +2} \\ \frac{f}{g}(2) = \frac{4}{22} \\ \frac{f}{g}(2) = \frac{2}{11}\)
The rectangle shown has a perimeter of 156 cm and the given area. Its length is 6 more than five times its width.
Write and solve a system of equations to find the dimensions of the rectangle.
Answers
l=6+5w
P=2(l+w)
156=2(6+5w+w)
156=2(6+6w)
156=12+12w
12+12w=156
12w=156-12
12w=144
w=12cm
l=6+5•12
l=6+60
l=66cm
Geometry 6.2
What is m∠N
Answers
Check the picture below.
the diagonals of an isosceles trapezoid are represented by x+10 and 2x+5.what is the value of x?
Answers
Answer:
x=5
Step-by-step explanation:
since it’s isosceles trapezoid, diagonals are equal.
x+10=2x+5
x=5
use green's theorem to find the counterclockwise circulation and outward flux for the field f=(7x−4y)i (9y−4x)j and curve c: the square bounded by x=0, x=4, y=0, y=4.
Answers
The counterclockwise circulation around c is 12 and the outward flux through c is zero.
Green's theorem is a useful tool for calculating the circulation and flux of a vector field around a closed curve in two-dimensional space.
In this case,
we have a field f=(7x−4y)i+(9y−4x)j and
a square curve c bounded by x=0, x=4, y=0, y=4.
To find the counterclockwise circulation, we can use the line integral of f along c, which is equal to the double integral of the curl of f over the region enclosed by c.
The curl of f is given by (0,0,3), so the line integral evaluates to 12.
To find the outward flux, we can use the double integral of the divergence of f over the same region, which is equal to zero since the divergence of f is also zero.
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Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. The cost per person is $30 minus $0.25 for every ticket sold. Gas and other miscellaneous costs are $200. Write the revenue and cost functions for this situation. Write the profit function for this situation. Find the number of tickets needed to maximize profit and give the maximum profit
Answers
Let's denote the number of tickets sold as .
The revenue function can be calculated by multiplying the number of tickets sold by the ticket price:
Revenue() = * (30 - 0.25)
The cost function consists of two components: the cost per person and the fixed costs for gas and miscellaneous expenses:
Cost() = * 30 - 0.25^2 + 200
To find the profit function, we subtract the cost function from the revenue function:
Profit() = Revenue() - Cost()
Profit() = * (30 - 0.25) - ( * 30 - 0.25^2 + 200)
Profit() = -0.25^2 + 30 - 0.25^2 + 200
To maximize profit, we need to find the value of that maximizes the profit function. This can be achieved by finding the vertex of the quadratic equation () = -0.25^2 + 30 + 200.
The x-coordinate of the vertex can be calculated using the formula = - / (2), where = -0.25 and = 30:
= -30 / (2 * -0.25)
= -30 / -0.5
= 60
Therefore, the number of tickets needed to maximize profit is 60.
To find the maximum profit, we substitute = 60 into the profit function:
Profit(60) = -0.25 * (60)^2 + 30 * 60 + 200
Profit(60) = -0.25 * 3600 + 1800 + 200
Profit(60) = -900 + 2000
Profit(60) = 1100
Hence, the maximum profit is $1100.
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there are 10 bags containing tickets numbered 1 to 20. ten students draw a ticket from each bag. one student drew tickets with the numbers: 5, 8, 18, 1, 14, 6, 8, 13, 8, 19 select the option that shows the correct mean, median, and mode.
Answers
The mean, median, and mode are:
Mean = 10
Median = 8
Mode = 8
To find the mean, median, and mode of the given set of numbers: 5, 8, 18, 1, 14, 6, 8, 13, 8, 19, we can perform the following calculations:
Mean: The mean is calculated by summing up all the numbers and dividing by the total count of numbers.
Mean = (5 + 8 + 18 + 1 + 14 + 6 + 8 + 13 + 8 + 19) / 10
= 100 / 10
= 10
Therefore, the mean is 10.
Median: The median is the middle value when the numbers are arranged in ascending order. If there are an odd number of values, the median is the middle number. If there are an even number of values, the median is the average of the two middle numbers.
Arranging the numbers in ascending order: 1, 5, 6, 8, 8, 8, 13, 14, 18, 19
Since there are 10 numbers, the median is the average of the two middle numbers, which are the 5th and 6th numbers:
Median = (8 + 8) / 2
= 16 / 2
= 8
Therefore, the median is 8.
Mode: The mode is the number(s) that appear(s) most frequently in the set.
In this case, the number 8 appears three times, which is more than any other number.
Therefore, the mode is 8.
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PLZ HELP URGENT !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
its 7. but try socratic
2k-3(2k-3)=45
step by step please?
Answers
Here’s the step by step solution
Answer:
k=-9
Step-by-step explanation:
\(2k-3(2k-3)=45\\\\2k-3(2k)-3(-3)=45\\\\2k-6k+9=45\\\\(2k-6k)+9=45\\\\(2-6)k+9=45\\\\-4k+9=45\\\\-4k+9-9=45-9\\\\-4k+0=36\\\\-4k=36\\\\\frac{-1}{4}(-4k) =\frac{-1}{4}(36) \\\\k=\frac{-36}{4} \\\\k=\frac{4*9}{4} \\\\k=\frac{9}{1} \\\\k=-9\)
Rewrite sin^2 (x) cos^2 (x) in expanded form, with no powers, parentheses or products.
Answers
cos^2(x) - cos^4(x)
sin^2 (x) cos^2 (x) can be written as sin^2 (x) cos^2 (x) = (1-cos^2(x)) cos^2(x)Expanding (1-cos^2(x)) cos^2(x) gives - cos^4(x) + cos^2(x)Therefore, sin^2 (x) cos^2 (x) in expanded form, with no powers, parentheses or products is cos^2(x) - cos^4(x).
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the miller school of business at ball state university claims to have a 73% graduate rate from its online mba program. a happy student believes that the 3-year graduation rate is higher than that. a sample of 500 students indicates that 371 graduated within three years. what is the lower limit for the 95% confidence interval for the true graduation rate? use decimals rather than percentages and round your answer to three decimal places.
Answers
The lower limit for the 95% confidence interval for the true graduation rate is 0.704.
What is confidence interval?The percentage (frequency) of acceptable confidence intervals that include the actual value of the unknown parameter is represented by the confidence level.
We can use the formula for the confidence interval for a proportion:
lower limit = sample proportion - z * √((sample proportion * (1 - sample proportion)) / sample size)
where z is the z-score corresponding to the desired level of confidence (95% in this case), and sample size is 500.
The sample proportion is 371/500 = 0.742.
The z-score for a 95% confidence level can be found using a standard normal distribution table or calculator, and is approximately 1.96.
Plugging in the values, we get:
lower limit = 0.742 - 1.96 * √((0.742 * 0.258) / 500)
lower limit = 0.704
Rounding to three decimal places, the lower limit for the 95% confidence interval for the true graduation rate is 0.704.
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What is -2-(-7)
Plsss explain
Answers
Answer:
14 I think I'm. it completely sure
Answer:
5
Step-by-step explanation:
you keep the sign of the first number
change the minus sign to addition
and the 7 to a positive
-2 +7
is the same as 7 - 2 if you switch the term
PLEASE HELP How can a company use a scatter plot to make future sale decisions
Answers
Answer:
From scatter plot companies can predict future sales, and what will happen next. To help with this predictions most companies draw a line through the scattered plot called best-fit line. This line should be close to most of the points on the scattered line. Approximately half the point on the top of the line and half on the bottom.In this case the company will ignore the points tat far away from the line.
Scatter plots are useful to compare two variables to see how they relate to one another (if there is any relationship at all). One example could be comparing the temperature outside versus the sales of ice cream. The general trend is that the warmer it gets, the more sales you'll have. So there's an upward trend. We can also say there's a positive correlation as both variables go up together (or go down together).
Contrast this with negative correlation where one variable goes up and the other goes down (eg: hours spent watching tv versus exam score).
Of course, the ice cream example could be too simple and often overused, so it might be better to use something more specific to the company in question. If you picked a company dealing with health/medicine, then you could look at something like height versus weight and see if there's a correlation going on.
Show how you would simplify the following expressions. Show your work.
Answers
1. 3 + (-5) = -2
4^ -2 = 1/4^2 = 1/16
2. -2+ (-4) = -6
3^-6 = 1/3^6 = 1/729
PLEASE HELP
in her stamp collection, Marta has 205 stamps . her cousin has 79 stamps. how many fewer stamps dose her cousin have?
Answers
Answer:
126 stamps
Step-by-step explanation:
205-79=126
Michael buy a baket of mangoe on ale for \$ 4$4 before tax.
The ale tax i 12%.
What i the total price Michael pay for the baket of mangoe?
Answers
Total price paid by Michael for the basket of mangoes is $4.48.
For the calculation of total price, price of sales tax will be added with selling price. Forming the equation -
Total price = 4 + 12%×4
Calculating the percentage of sales tax on Right Hand Side of the equation
Total price = 4 + 12×4/100
Solving the percentage part on Right Hand Side of the equation
Total price = 4 + 0.48
Performing addition on Right Hand Side of the equation
Total price = $4.48
Hence, Michael has to pay $4.48 after addition of sales tax for the basket of mangoes.
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Learn more
The chance of winning based on the type of lottery tickets we purchase is 0.05, or 1 in 20. We will play each Saturday until we win. What is the probability of winning on the fourth ticket
Answers
The probability of winning on the fourth ticket is 0.83%.
Given, The probability of winning based on the type of lottery tickets we purchase is 0.05 or 1 in 20.
This means that the probability of winning is 1/20=0.05.We will play each Saturday until we win.
This statement gives us a hint that we need to find the probability of winning after four attempts/tries
Let the probability of not winning be (1-0.05)=0.95.
Now, the probability of not winning after 4 tries is:P(N) = 0.95 x 0.95 x 0.95 x 0.95 = 0.8145
The probability of winning on the fourth ticket is equal to the probability of not winning in three tickets (0.8145) multiplied by the probability of winning on the fourth ticket (0.05).
Thus, the probability of winning on the fourth ticket is:0.8145 x 0.05 = 0.040725 or 0.04
Summary: The probability of winning based on the type of lottery tickets we purchase is 0.05 or 1 in 20. We will play each Saturday until we win. The probability of winning on the fourth ticket is 0.83%.
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PLEASE HELP ME IN MATHHH PLS
Answers
Answer:-83/25 and 27/40
Type the correct answer in the box. Use numerals instead of words. If necessary, use/ for the fraction bar.
Given the figure, find the total area of the shaded region.
D
8-
6-
4-
2-
O
-2-
o
S
The area of the shaded region is
B
R
8
с
square units
Answers
The value of the total area of the shaded region are,
⇒ 42 units²
We have to given that;
Sides of rectangle are,
AB = 9
BC = 6
Hence, The area of rectangle is,
⇒ 9 x 6
⇒ 54 units²
And, Area of triangle is,
A = 1/2 × 4 × 6
A = 12 units²
Thus, The value of the total area of the shaded region are,
⇒ 54 - 12
⇒ 42 units²
So, The value of the total area of the shaded region are,
⇒ 42 units²
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A pregnant woman with a past history of a cesarean section is at GREATEST risk for: Group of answer choices breech birth. preeclampsia. uterine rupture. prolapsed cord.
Answers
A pregnant woman with a past history of a cesarean section is at the greatest risk for uterine rupture.
Uterine rupture refers to a tear or separation of the uterine wall, which can be a life-threatening complication during pregnancy or labor. Women who have had a previous cesarean section are at an increased risk of uterine rupture compared to those who have not undergone the procedure. This is because the scar tissue from the previous incision can weaken the uterine wall, making it more susceptible to rupture during subsequent pregnancies. The risk of uterine rupture is highest during labor, especially if the woman attempts a vaginal birth after cesarean (VBAC). The pressure and contractions of labor can place stress on the scarred area, increasing the chances of a uterine tear. Uterine rupture can lead to severe bleeding, fetal distress, and other complications that may require emergency medical intervention, including a cesarean delivery. It is important for pregnant women with a history of cesarean section to work closely with their healthcare providers to assess the risks and make informed decisions about their mode of delivery. This may involve considering the individual's specific circumstances, such as the type of uterine incision from the previous cesarean and any additional factors that may increase the risk of uterine rupture.
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person sold a house
for Rs 20,00,000 then he gets 5% commission. Find
commission amount .
Answers
Answer:
₹1, 00,000
Step-by-step explanation:
Amount of commission
\( = 5 \% \: of \: 20, 00, 000 \\ \\ = \frac{5}{100} \times 20, 00, 000 \\ = 5 \times 20000 \\ = Rs. \: 1, 00, 000\)
How do you solve this equation
-1 = -1 + (p/8)
Answers
The solution to the given equation -1 = -1 + (p/8) is p = 0.
What is the solution to the given equation?Given the data in the question;
-1 = -1 + (p/8)p = ?Solve for p to determine the solution to the equation.
-1 = -1 + (p/8)
Rewrite the equation as;
-1 + (p/8) = -1
Next, move all terms containing p to the left side and all constants to the right side of the equation.
-1 + (p/8) = -1
(p/8) = -1 + 1
Add -1 and 1
p/8 = -1 + 1
p/8 = 0
p = 0 × 8
p = 0
The solution to the given equation -1 = -1 + (p/8) is p = 0.
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Solve each trigonometric function IN THE INTERVAL [0, 2).
sin(x) + 1 = cos(x)
Answers
The solution of the trigonometric function sin(x) + 1 = cos(x) in the interval of [0, 2) will be 0.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
The trigonometric function is given below.
sin(x) + 1 = cos(x)
Simplify the equation, then we have
sin(x) + 1 = cos(x)
(sin(x) + 1)² = cos² (x)
(sin(x) + 1)² = 1 - sin² (x)
sin² (x) + 2sin x + 1 = 1 - sin² (x)
2sin² (x) = 0
sin (x) = sin 0
x = 0
The solution of the trigonometric function sin(x) + 1 = cos(x) in the interval of [0, 2) will be 0.
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Course activity: modeling with functions
Monica and Shanna are saving the money they earn from their after-school jobs to go on a school trip. Monica starts with $20 and then saves $14 each week. Shanna's savings are shown in the table.
Monica’s savings can he modeled by a ____ table
Shanna’s savings can be modeled by a ____ table
Answers
Answer:Both A and C
Step-by-step explanation:
Four friends want to take a vacation together, so each one gets a part-time job. Each person has 8 weeks to save $720 for the vacation. Analyze the four individual plans below and decide which of the four people will reach his or her goal of saving $720 for vacation.
Friend A: Works 13 hours per week at $6.95 per hour.
Friend B: Works 15 hours per week at $5.85 per hour.
Friend C: Works 18 hours per week at $5.25 per hour.
Friend D: Works 11 hours per week at $7.80 per hour. there!
This is my hw. heelp
Answers
Answer:
1/14
Step-by-step explanation: