Answers
Take the Laplace transform of both sides:
L[y'' - 4y' + 8y] = L[δ(t - 1)]
I'll denote the Laplace transform of y = y(t) by Y = Y(s). Solve for Y :
(s²Y - s y(0) - y'(0)) - 4 (sY - y(0)) + 8Y = exp(-s) L[δ(t)]
s²Y - 4sY + 8Y = exp(-s)
(s² - 4s + 8) Y = exp(-s)
Y = exp(-s) / (s² - 4s + 8)
and complete the square in the denominator,
Y = exp(-s) / ((s - 2)^2 + 4)
Recall that
L⁻¹[F(s - c)] = exp(ct) f(t)
In order to apply this property, we multiply Y by exp(2)/exp(2), so that
Y = exp(-2) • exp(-s) exp(2) / ((s - 2)² + 4)
Y = exp(-2) • exp(-s + 2) / ((s - 2)² + 4)
Y = exp(-2) • exp(-(s - 2)) / ((s - 2)² + 4)
Then taking the inverse transform, we have
L⁻¹[Y] = exp(-2) L⁻¹[exp(-(s - 2)) / ((s - 2)² + 4)]
L⁻¹[Y] = exp(-2) exp(2t) L⁻¹[exp(-s) / (s² + 4)]
L⁻¹[Y] = exp(2t - 2) L⁻¹[exp(-s) / (s² + 4)]
Next, we recall another property,
L⁻¹[exp(-cs) F(s)] = u(t - c) f(t - c)
where F is the Laplace transform of f, and u(t) is the unit step function
[tex]u(t) = \begin{cases}1 & \text{if }t \ge 0 \\ 0 & \text{if }t < 0\end{cases}[/tex]
To apply this property, we first identify c = 1 and F(s) = 1/(s² + 4), whose inverse transform is
L⁻¹[F(s)] = 1/2 L⁻¹[2/(s² + 2²)] = 1/2 sin(2t)
Then we find
L⁻¹[Y] = exp(2t - 2) u(t - 1) • 1/2 sin(2 (t - 1))
and so we end up with
y = 1/2 exp(2t - 2) u(t - 1) sin(2t - 2)
Related Questions
From a group of 15 men and 12 women, how many ways can you select a group of 2?
Answer:
Answers
The total number of ways this can be done together is 105 × 66 = 6930
This is a combination question question it has to do with selection.
In order to group 2 men from 15 men, this can be expressed using the combination formula as shown;
15C2 = 15!/(15-2)!2!
15C2 = 15!/13!2!
15C2 = 15×14/2
15C2 = 105 ways
Similarly in order to group 2 women from 12 men, this can be expressed using the combination formula as shown;
12C2 = 12!/(12-2)!2!
12C2 = 12!/10!2!
12C2 = 12×11/2
12C2 = 66 ways
The total number of ways this can be done together is 105 × 66 = 6930
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Find the product. Simplify your answer.
(m+2)(3m+2)
Answers
there you go the answer is in the picture
3m^2+8m+4
A fire is sighted due west of lookout A. The bearing of the fire from lookout B, 8.6 miles due south of A, is N 42°26'W. How far is the fire from B (to the nearest tenth of a mile)? 7) A)13.7 mi B)11.7 mi C)12.7 mi D)14.7 mi
Answers
Answer:
11.7 miles
Step-by-step explanation:
I hope this helped!
A scale model of a bicycle has a scale of 1 : 25. The real bicycle has a length of 1.75 m. What is the length of the model in cm?
Answers
Answer:
It's 7 I just did this question today and got it right
In the penny game exercise, why do you think it takes longer for one person to flip five batches of two pennies then one batch of ten pennies
Answers
It takes longer for one person to flip five batches of two pennies than one batch of ten pennies, mainly for a simple reason: although 5 x 2 is equal to 10, that is, it will take the same time to flip the coins, 5 batches is greater than 1 batch, which will take longer to manipulate each of these, increasing the total time.
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Please helpppp I will mark you branliest pleaseee
Answers
Answer:
no
Step-by-step explanation:
the ratios shown in this table are not proportionate. You will also never get (0,0) in this table.
The area of a rectangle is 24 square inches. Determine both dimensions if the length is 6 inches less than three times the width
Answers
Answer:
width is 4 inches
length is 6 inches
Step-by-step explanation:
the length is 6 inches less than three times the width
l = 3w - 6
a = l * w
a = (3w - 6)w
a = 3w^2 - 6w
area of the rectangle is 24 square inches
3w^2 - 6w = 24
subtract 24 from both sides
3w^2 - 6w - 24 = 0
Divide both sides by 3
w^2 - 2w - 8 = 0
Factor
(w - 4)(w + 2) = 0
w = {-2, 4}
disregard the negative root since measurement must be positive
w = 4
l = 3w - 6
l = 3(4) - 6
l = 6
WILL GIVE BRAINLIEST ‼️ The graph represents the piecewise function:
Answers
Answer:
[tex]f(x) = \left \{ {{x + 3} \quad \text{if} \ -3 \leq x < -1\atop {5} \quad \text{if} \ -1 \leq x \leq1} \right.[/tex]
Step-by-step explanation:
You can find the first function if you know the general form of a line:
[tex]y = mx + b[/tex]
where [tex]m[/tex] is the slope and [tex]b[/tex] the intercept point with the [tex]y[/tex]-axis.
So in the first case the slope is 1 because the next formula
[tex]m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2-0}{-1+3} = 1[/tex]
And you can find the interception point if extend the line, then in our case is 3.
So the function is:
[tex]f(x) = x + 5[/tex]
In the other case is constant function that's mean that the function has the next form
[tex]f(x) = k[/tex]
Where [tex]k[/tex] is a constant.
So you only have to observe what is the value for each point in the interval, in this occasion is 5.
So the final answer is:
[tex]f(x) = \left \{ {{x + 3} \quad \text{if} \ -3 \leq x < -1\atop {5} \quad \text{if} \ -1 \leq x \leq1} \right.[/tex]
Does anyone know this?
Answers
Answer:
I think 7.6 i remember i had something like this
Step-by-step explanation:
20 feel from the ground i need a ladder placed 15 feet from the house what is the length
Answers
Answer:
helooo
Step-by-p explanation:
dur rn help pls ))))))))))
Answers
I believe it is a because I had a question just like that in 8th grade
Problem example: For the given sequence: -2,2,6,10,14,18
Write a recursive formula to find the nth term: a = an-1+4
Write a explicit formula to find the nth term: a = -2 + 4(n-1)
What is the 64th term in the sequence?
Answers
Answer:
250
Step-by-step explanation:
You can take the explicit formula and simply substitute the n value with 64:
-2 + 4 * (64 - 1)
-2 + 4 * (63)
-2 + 252
250
Ms. McCullough bought a bunch of bananas. If she ate 2/8
the bananas and Mrs. Byrd ate 1/4, what fraction of bananas
remains?
Answers
Answer:
1/2
Step-by-step explanation:
that's the answer because 2/8-1/4 is 1/2
Are the two ratios that make up a proportion
always, sometimes, or never equivalent
Answers
Answer:
Always
Step-by-step explanation:
In mathematics, when we say that two ratios form a proportion, we mean that the two ratios are equal.
Ex: Ratios and Proportions - Proportions - In Depth. A proportion is simply a statement that two ratios are equal. It can be written in two ways: as two equal fractions a/b = c/d; or using a colon, a:b = c:d. The following proportion is read as "twenty is to twenty-five as four is to five."
A proportion is two ratios which have been set equal to each other; a proportion is an equation that can be solved. When I say that a proportion is two ratios that are equal to each other, I mean this in the sense of two fractions being equal to each other. For instance, 105 equals 21 .
The two ratios sometimes that make up a proportion.
What is Proportional?Any relationship that is always in the same ratio and quantity which vary directly with each other is called the proportional.
We know that;
Two ratios that make up a proportion when both ratios are equal to each other.
And, When two ratios are not equal then it cannot make a proportion.
Hence, The two ratios sometimes that make up a proportion.
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In the past, Marcie's father rode his bike 108 miles in 7.5 hours. Her mother rode the same distance in 8 hours. Marcie plans to ride her bike 108 miles at a steady rate of 18 mph for y hours. Will she match her father or mothers time? Use the equation 108÷y=18 to justify your answer.
Answers
Using the speed - distance relationship, Marcie's time will be 6 hours. Hence, she'll match her parents time.
Recall :
Speed = distance / timeTo obtain Marcie's time :
18 = 108 / yWe can solve for y thus :
y × 18 = 108
y = 108 / 18
y = 6 hours
Since, Marcie's will cover the same distance in 6 hours, the she'll match and surpass her parents time.
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help me due today it is math
Answers
Answer:
2
Step-by-step explanation:
sum of digits=1+2+5+8=16
smallest term to be added to make it divisible by 9 is 2
∵16+2=18÷9=2
write the value of the underlined digit in words 6,302,450? 3 being the underlined number.
Answers
Answer:
Three hundred thousand
Step-by-step explanation:
Hope this helps
Alvin’s family used 18.5 gallons of gas to drive 518 miles. How many miles did they drive per gallon of gas?
Please help me
Answers
Answer:
the answer is 1.2
Step-by-step explanation:
The number of miles Alvin’s family drive per gallon of gas is 28 miles.
Given that, Alvin’s family used 18.5 gallons of gas to drive 518 miles.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Number of miles drive per gallon of gas = Total number of miles ÷ Total number of gallons
= 518 ÷ 18.5
= 28 miles
Hence, the number of miles Alvin’s family drive per gallon of gas is 28 miles.
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A package contains 12 ounces of cereal .If you use 1 portion of this cereal which is 1 1/3 how much would be left in the cereal box?
Answers
To answer this question, you would need to divide the total amount of cereal in the box by the serving size.
This would be 12 2/3 divided by 1 1/3.
We can write each mixed number as an improper fraction first before we divide:12 2/3 = 38/3 and 1 1/3 = 4/3
Now divide 38/3 by 4/3. Recall that dividing by a fraction is accomplished by multiplying by the reciprocal of the divisor: 38/3 * 3/4 = 114/12 = 38/4 OR 19/2
Also, 19/2 is equivalent to 9 and 1/2. There would be 9 and 1/2 servings in the box of cereal.
Hope this helps
-----------------------------
Answers
Answer:
thanks for tha points
Step-by-step explanation:
solve the equation 5/x+3 + 4/x+2 = 2
Answers
Answer:
5 + 4 = 9
x × x = X2
3 + 2 = 5
9/x + 5 = 2
Answer:
5/x+3+4/x+2=2
=9/x+5=2
9/x=-3
Multiply both sides by x
9= -3x
Divide by -3
Answer x= -3
Hope this helps
what is the distance between the coordinates (9, -2) and (9, 8) show your work on a coordinate grid pleas help mee
Answers
Answer:
10
Formular for finding distance between two points; (x2-x1)² + (y2-y1)² (all square root)
X1=9
X2=9
Y1=-2
Y2=8
Distance= (9-9)²+(-2-8)² (all square root)
=(0)²+(-10)² (all square root)
=0 + 100 (square root)
=100 square root
=10
-2.875 greater than or less than -2 7/8
Answers
Answer:
They are both the same.
Step-by-step explanation:
-2.875
7/8 = 0.875 thus -2 7/8 is equal to -2.875
solve for x and y then find the measure of each angle
Answers
Step-by-step explanation:
∵ MP is a straight line.
∴ ∠MNQ + ∠PNQ = 180°
[tex](3y + 8)° + (3y + 22)° = 180° \\ 3y + 8 + 3y + 22 = 180 \\ 6y + 30 = 180 \\ 6y = 150 \\ y = 25[/tex]
∵ MP is a straight line.
∴ ∠MNR + ∠PNR = 180°
[tex](2x + 5)° + (3x - 55)° = 180° \\ 2x + 5 + 3x - 55 = 180 \\ 5x - 50 = 180 \\ 5x = 230 \\ x = 46[/tex]
if there was a group of ten people the probability of them being born on the same day of the same year is 1, true or false and why
Answers
Answer:
True
Step-by-step explanation:
For instance, twins are born on the same day but imagine ten people instead of two being born on the same day.
Please help I do not understand thanks
Answers
Three times the difference “2x minus 1” is the opposite of 9 times x. Solve for x.
Answers
Answer:
x = 1/5 - first box
Step-by-step explanation:
Let's interpret this by using what I call the "isolated-x method."
First, we can simplify whatever's being affected in the parentheses.
#1: 3 times (2x) is equal to 6x
#2: 3 times (- 1 | equal to -1) results in -3, or + -3
So now we're working with 6x - 3 = -9x.
Now, we'll add a positive 9x to the -9x, since it can be hugely helpful to have one side result in 0.
Now we're working with 15x (6x + our new 9x) - 3 = 0
Since we have "whole like terms" we can now easily add 3 to each side.
Now we're working with 15x = 3 (0 + our new 3)
We can finish by dividing each side by 3.
15x = 3 now turns into 5x = 1.
To get from 5 to 1 you can multiply 5 by its "multiplicative inverse," also known as 1/5.
Therefore, x = 1/5
Hopefully this helped a bunch! Tell me if you need any further assistance...
( :
Strawberries cost $3.00 per pound. For each question
a.
How many pounds of strawberries can you buy for $1.00?
b.
How many pounds of strawberries can you buy for $11.0
Answers
Answer:
A. 1/3 pound
B. 11/3 or 3.666666667
Step-by-step explanation:
:)
A secured loan would most likely be used to pay for
A. college
B. a car
C. groceries
D. a phone
Answers
A secured loan would most likely be used to pay for a car, the correct option is B.
We are given that;
The four options
Now,
A secured loan is a loan that is backed up by collateral pledged by the borrower, such as a car or property.
The lender can sell the collateral to cover repayment of the loan if the borrower defaults.
Car, since it is a common type of collateral for secured loans.
College, groceries and a phone are less likely to be used as collateral for secured loans, since they are either intangible, perishable or low-value assets.
Therefore, by loan the answer will be car.
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What is 3.59 rounded to the nearest whole number?
Answers
Answer:
its 4 ezy
Step-by-step explanation:
please mark as brainlists
