Use the Laplace transform to solve the given initial-value problem.
y' + 3y = f(t), y(0) = 0
where f(t) = t, 0 ≤ t < 1 0, t ≥ 1
Answers
Answer:
The solution to the given Initial - Value - Problem is \(y(t) = \frac{-1}{9} + \frac{1}{3}t + \frac{1}{9}e^{-3t} - [\frac{-1}{9} + \frac{1}{3}t - \frac{2}{9}e^{-3(t-1)}]u(t-1)\)
Step-by-step explanation:
y' + 3y = f(t).................(1)
f(t) = t when 0 ≤ t < 1
f(t) = 0 when t ≥ 1
Step 1: Take the Laplace transform of the LHS of equation (1)
That is L(y' + 3y) = sY(s) + 3Y(s) = Y(s)[s + 3]..............(*)
Step 2: Get an expression for f(t)
For f(t) = t when 0 ≤ t < 1
f₁(t) = t (1 - u(t - 1)) ( there is a time shift of the unit step)
For f(t) = 0 when t ≥ 1
f₂(t) = 0(u(t-1))
f(t) = f₁(t) + f₂(t)
f(t) = t - t u(t-1)................(2)
Step 3: Taking the Laplace transform of equation (2)
\(F(s) = \frac{1}{s^2} - e^{-s} ( \frac{1}{s^2} + \frac{1}{s})\)...............(**)
Step 4: Equating * and **
\(Y(s) [s + 3]=\frac{1}{s^2} - e^{-s} ( \frac{1}{s^2} + \frac{1}{s}) \\Y(s) = \frac{1}{s^2(s+3)} - e^{-s} ( \frac{1}{s^2(s+3)} + \frac{1}{s(s+3)})\).......................(3)
Since y(t) is the solution we are looking for we need to find the Inverse Laplace Transform of equation (3) by first breaking every fraction into partial fraction:
\(\frac{1}{s^2 (s+3)} = \frac{-1}{9s} + \frac{1}{3s^2} + \frac{1}{9(s+3)}\)
\(\frac{1}{s (s+3)} = \frac{1}{3s} + \frac{1}{3(s+3)}\)
We can rewrite equation (3) by representing the fractions by their partial fractions.
\(Y(s) = \frac{-1}{9s} + \frac{1}{3s^2} + \frac{1}{9(s+3)} - e^{-s} [\frac{-1}{9s} + \frac{1}{3s^2} + \frac{1}{9(s+3)} + \frac{1}{3s} + \frac{1}{3(s+3)}]\\Y(s) = \frac{-1}{9s} + \frac{1}{3s^2} + \frac{1}{9(s+3)} - e^{-s}[\frac{2}{9s} + \frac{1}{3s^2} - \frac{2}{9(s+3)}]\)................(4)
step 5: Take the inverse Laplace transform of equation (4)
\(y(t) = \frac{-1}{9} + \frac{1}{3}t + \frac{1}{9}e^{-3t} - u(t-1)[\frac{2}{9} + \frac{1}{3}(t-1) - \frac{2}{9}e^{-3(t-1)}]\)
Simplifying the above equation:
\(y(t) = \frac{-1}{9} + \frac{1}{3}t + \frac{1}{9}e^{-3t} - [\frac{-1}{9} + \frac{1}{3}t - \frac{2}{9}e^{-3(t-1)}]u(t-1)\)
The Laplace transform is use to solve the differential equation problem.
The solution for the given initial-value problem is,
\(y(t)=\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{1}{9}e^-3t-\left[\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{2}{9}e^{-3(t-1)}\right]u(t-1)\)
Given:
The given initial value problem is \(y' + 3y = f(t)\).
Consider the left hand side of the given equation.
\(y'+3y\)
Take the Laplace transform.
\(L(y' + 3y) = sY(s) + 3Y(s) \\L(y' + 3y) = Y(s)[s + 3]\)
Consider the right hand side and get the expression for \(f(t)\).
\(f(t) = t\) when 0 ≤ t < 1
From time shift of the unit step
\(f_1(t) = t (1 - u(t - 1))\)
For f(t) = 0 when t ≥ 1
Now,
\(f_2(t) = 0(u(t-1))f(t) = f_1(t) + f_2(t)f(t) = t - t u(t-1)\)
Take the Laplace for above expression.
\(F(s)=\dfrac{1}{s^2}-e^{-s}\left(\dfrac{1}{s^2}+\dfrac{1}{s}\right)\)
Now, the equate the above two equation.
\(Y(s)\left[s+3\right ]=\dfrac{1}{s^2}-e^{-s}\left(\dfrac{1}{s^2}+\dfrac{1}{s}\right)\\Y(s)=\dfrac {1}{(s^2(s+3))}-e^{-s}\left(\dfrac{1}{(s^2(s+3))}+\dfrac{1}{s(s+3)\right)}\)
Find the inverse Laplace for the above equation.
\(\dfrac{1}{(s^2(s+3))}=\dfrac{-1}{9s}+\dfrac{1}{3s^2}+\dfrac{1}{9(s+3)}\\\dfrac{1}{(s(s+3))}=\dfrac{1}{3s}+\dfrac{1}{3(s+3)}\)
Calculate the partial fraction of above equation.
\(Y(s)=\dfrac{-1}{9s}+\dfrac{1}{3s^2}+\dfrac{1}{9(s+3)}-e^{-s}\left[\dfrac{-1}{9s}+\dfrac{1}{3s^2}+\dfrac{1}{9(s+3)}+\dfrac{1}{3s}+\dfrac{1}{3(s+3)}\right]\\Y(s)=\dfrac{2}{9s}+\dfrac{1}{3s^2}+\dfrac{1}{9(s+3)}-e^{-s}\left[\dfrac{2}{9s}+\dfrac{1}{3s^2}-\dfrac{2}{9(s+3)}\right]\)
Take the inverse Laplace of the above equation.
\(y(t)=\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{1}{9}e^-3t-\left[\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{2}{9}e^{-3(t-1)}\right]u(t-1)\)
Thus, the solution for the given initial-value problem is,
\(y(t)=\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{1}{9}e^-3t-\left[\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{2}{9}e^{-3(t-1)}\right]u(t-1)\)
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Related Questions
You lease a car at $23,495 for 3 years at $429.95 a month with a $500 down payment. The interest is 30% of the payments and $4,643.46 in interest is paid over 3 years. What is the remaining balance when the lease ends? How did you arrive at $12,160.26?
Answers
Answer:
Step-by-step explanation:
total interest paid is given as $4,643.46.
total payments = $429.95 x 36 months = $15,478.20
total lease payments = total payments - total interest
total lease payments = $15,478.20 - $4,643.46 = $10,834.74
Remaining balance = Total cost of the lease - Total lease payments
$23,495 - $10,834.74 = $12,660.26
Match the following. Match the items in the left column to the items in the right column. 1. domain the first element of a relation or function; also known as the input value. 2. output a relation in which every input value has exactly one output value. 3. input the x-value of a function. 4. relation any set of ordered pairs (x, y) that are able to be graphed on a coordinate plane. 5. function the y-value of a function. 6. range the second element of a relation or function; also known as the output value.
Answers
The matching of items and their corresponding descriptions are 1. Domain, 2.Output, 3. Input, 4. Relation, 5. Function, and 6. Range.
What is the appropriate matching of the following items?1. Domain - the first element of a relation or function; also known as the input value.
3. Input - the x-value of a function.
6. Range - the second element of a relation or function; also known as the output value.
4. Relation - any set of ordered pairs (x, y) that are able to be graphed on a coordinate plane.
2. Output - a relation in which every input value has exactly one output value.
5. Function - the y-value of a function.
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dose the order in which you add -9 and 9 matter? Explain your reasoning.
Answers
Answer:
Not particularly, as your answer would be the same either way, though practicality would turn negative 9 into minus 9, so it would go from
x + (-9) + 9 and become x -9 +9.
Also, the -9 and the 9 would cancel out.
There are 9 roses in a vase of 14 flowers. The rest are daisies. (Plz look at picture and no wrong answers please I need help)
Answers
Answer:
flowers : daises
14 : 5
roses : daises
9 : 5
Step-by-step explanation:
14 - 9 = 5, so there are 5 daisies
flowers : daises
14 : 5
roses : daises
9 : 5
Answer:
a) flowers:daisies = 14:5
b) roses:daisies = 9:5
Step-by-step explanation:
A sum of money is to be divided among Anthea, Barry and Carol in the ratio 2:5:1
respectively. If Barry got $200 more than Carol, calculate:
The sum of money?
Answers
Answer:
$400
Step-by-step explanation:
Lets money that Anthea got be 2x
than Barry's money be 5x and Carol's be x
Barry got $200 more than Carol
This means that 5x is 200 more than x
5x = x + 200
4x = 200
x = 50
The sum of money is 2x + 5x + x
so this is 8x
8x = 50 * 8 = 400
What is the total length of border needed in feet and
inches?
Answers
7x - 2 = 9x+5
Please help!!
Answers
Answer:
x=-7/2
Step-by-step explanation:
Answer:
x = -7/2 or -3.5
Step-by-step explanation:
7x - 2 = 9x + 5
Add 2 to both sides
7x = 9x + 7
Subtract 9x from both sides
-2x = 7
Divide both sides by -2
x = -7/2
x = -3.5
what is the surface area of 10 in 10 in 10in
Answers
The surface area of a three dimensional shape is the total area of all of the faces. To find the surface area of a shape, we find the area of each face and add them together. E.g. The measurements here are in centimetres so the surface area is measured in square centimetres ( c m 2 ) (cm^2) (cm2).
On your w-2 it states that you paid $12,042 in federal taxes. When filing your taxes and after all deductions, it says you should have paid $10,628 in federal taxes. How much is your tax refund? Please round to the nearest cent and do not put a dollar sign or spaces in your answer.
Answers
The amount of your tax refund is $1,414
how to determine how much is your tax refundTo find the tax refund, we need to subtract the amount of federal taxes owed after deductions from the amount of federal taxes already paid.
Tax refund = Federal taxes paid - Federal taxes owed after deductions
Tax refund = $12,042 - $10,628
Tax refund = $1,414
Rounding to the nearest cent, the tax refund is $1,414.00.
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why are inverse realashinships between operations used to solve twop step inqualites
Answers
Inverse relationships between operations are used to solve two-step inequalities because they allow us to isolate the variable on one side of the inequality.
A two-step inequality involves two operations that need to be undone in reverse order to solve for the variable. Inverse relationships between operations are used to "undo" these operations, leading to an isolated variable.
For example, consider the inequality 2x + 5 > 11. The first operation being performed on x is multiplication by 2, and the second operation is adding 5. To solve for x, we need to undo these operations in reverse order.
The inverse relationship of multiplication by 2 is division by 2, and the inverse relationship of adding 5 is subtracting 5. Therefore, we can solve the inequality as follows:
2x + 5 > 11
2x > 6 (subtract 5 from both sides)
x > 3 (divide both sides by 2)
Here, we first subtracted 5 from both sides to undo the addition of 5, and then divided both sides by 2 to undo the multiplication by 2.
Inverse relationships between operations are used in two-step inequalities because they help to simplify the equation by undoing the operations one by one, leading to an isolated variable.
By using inverse relationships between operations, we can efficiently and systematically solve two-step inequalities and isolate the variable to find the solution set.
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Draw an angle in standard position with the given measure. 8/3 π
Answers
The asterisk (*) represents the vertex of the angle, the initial side is the positive x-axis, and the terminal side is the red line segment that forms an angle of 8/3 π with the positive x-axis.
What is the standard position of angles?
Standard position simply means that the vertex of the angle is at the origin of the circle and that one ray of the angle is on the positive x-axis. The other ray of the angle is placed at the angle measure formed by traveling counter-clockwise along the circle.
To draw an angle in standard position with measure 8/3 π, follow these steps:
Start with the positive x-axis.
Counterclockwise, rotate the initial side of the angle until it coincides with the positive x-axis.
Starting from the initial side on the positive x-axis, rotate the terminal side of the angle in a counterclockwise direction until the angle measures 8/3 π.
The resulting angle will have a measure of 8/3 π and will look like the following:
|
|
--------*-----------
|
|
Here, the asterisk (*) represents the vertex of the angle, the initial side is the positive x-axis, and the terminal side is the red line segment that forms an angle of 8/3 π with the positive x-axis.
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Use your knowledge of angle relationships to solve for x in the transversals below. Then find the measure of each identified angle.
Your answer should be THE MEASURE OF THE ANGLE!!
Angle=
Answers
Hope it helps :))))) .....
What type of association does the graph show?
A. positive nonlinear
B. positive linear
C. negative nonlinear
D. negative linear
Answers
Answer:
B. Positive linear
Step-by-step explanation:
First, this graph is a linear graph because a linear graph is a straight line. The graph in the diagram is also a straight line, so it is linear.
Also, notice how as x increases, y increases. This means that the graph is positive
15 Points!! need help❤️
Answers
Answer:
70 degrees
Step-by-step explanation:
The measure of the central angle equal the measure of the arc it intercepts.
So arc BC is 70 degrees.
A right triangle. The length of the hypotenuse is 13 units . The length of one leg is 5 units. The length of the other leg is b units
Answers
Answer:
approximately 13
Step-by-step explanation:
Answer:
The answer is D
Step-by-step explanation:
Hope this helps :)
A taxi company charges $9 for pick-up plus $0.35 for each mile. Select the expressions that represent the cost in dollars for a taxi pick up and m miles. Select all that apply.
A. 9+0.35m
B. 9m+ 0.35
C. 0.35m+9
D. The product of 9 and m plus 0.35
Please answer quick
Answers
_____consists of activities that are
not required as part of one's formal
role in the organization
Select one:
a. None of the above
b. Lobbying
c. Influence
d. Political behaviour
= Political behaviour
Answers
Step-by-step explanation:
political behaviour consists of activities that are
not required as part of one's formal
role in the organization
Gabriela drank three fourths of her glass of milk and jenny drank six eighths of her glass of milk if the glasses were the same size did they drink the same amount ? Use cross multiplication to prove your answer
Answers
Answer:
just croos multi
Step-by-step explanation:
its very easy
In trapezoid PQRS, the lengths marked
are in feet. What is the area of the
trapezoid in square feet?
S
5
3
P 10
F.
44
G. 50
H. 62.5
J. 88
K. 100 dion
12
Q
R
Answers
The areas of the trapezoid is 50 units².
How to find the area of a trapezoid?The trapezoid PQRS have bases 10 and 12 units respectively. The height of the trapezoid is unknown. Let's find the area of the trapezoid as follows:
area of trapezoid = 1 / 2 (a + b)h
where
a and b are the basesh = height of the trapezoidTherefore,
a = 10 units
b = 15 units
Using Pythagoras's theorem,
5² - 3² = h²
h = √25 - 9
h = √16
h = 4 units
area of trapezoid = 1 / 2 (10 + 15)4
area of trapezoid = 1 / 2 × 25 × 4
area of trapezoid = 100 / 2
area of trapezoid = 50 units²
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plz solve it CORRECTLY step by step plz
Answers
Answer:
Step-by-step explanation:
area of circle=π\(r^{2}\)
=22/7*\(21^{2}\)
=22/7*441
=9702/7
=1386 \(cm^{2}\)
Answer:
1386cm sq
Step-by-step explanation:
formula to find area of circle is 22/7*radius sq
which is 22/7 * 21 * 21
= 22 * 3 * 21
= 1386
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A square garden has a length of (x+3) ft and a width of (x+2) ft. what is the perimeter and area of the garden?
Answers
Answer:
Perimeter:\(4x+10\) feet
Area:\(x^{2}+5x+6\) feet
Step-by-step explanation:
The perimeter is equal to 2*width +2*length. The width is x+2 and the length is x+3, therefore the perimeter is equal to 2x+4+2x+6 which equals 4x+10.
The area is equal to width*length
(x+3)(x+2)=\(x^{2}+2x+3x+6=x^{2}+5x+6\)
what is 2043.666666 rounded to 2 decimal places
Answers
Answer:
\(2043.67\)
Step-by-step explanation:
Hundredths is at 2 decimal places.
The thousandths place is higher than 5, so add 1 to the hundredths place.
Answer:
2043.67
Step-by-step explanation:
If you’ve ever rounded a number, you would know that if it’s 5 or higher, round it up, and if it’s 4 or lower, round it down. In this case, the second decimal place reads ’6’ which is higher that 5, so we round up. The rest of the numbers stay the same
2043.67
The length of a rectangle is 4 yd longer than its width. If the perimeter of the rectangle is 40 yd, find its area.
Answers
Answer:
96
Step-by-step explanation:
A random sample of 300 students is selected from a large group of students who use a computer-equipped classroom on a regular basis. Occasionally, students leave their USB drive in a computer. Of the 300 students questioned, 180 said that they write their name on their USB drive. Which of the following is a 98 percent confidence interval for the population of all students using the classroom who write their name on their USB drive? 0.4 + 2.331 (0.4)(0.6) 300 (0.4)(0.6) 300 0.4 - 1.967 0.6 +2.05. (0.6)(0.4) 300 90.6. + 2.331 (0.6)(0.4) 300 0.6 I 1.96V (0.6)(0.4) 300
Answers
The 98 percent confidence interval for the population of all students using the classroom who write their name on their USB drive is B) 0.4 ± 1.967 (0.4)(0.6) / √300.
The sample proportion (p-hat) of students who write their name on their USB drive is 0.4, based on 180 out of 300 students. The sample size is large enough (n > 30) to use the normal approximation to the binomial distribution to calculate the confidence interval.
The standard error of the sample proportion is (p(1-p)/n)^0.5, where p is the population proportion and n is the sample size.
The confidence interval is calculated by adding and subtracting the critical value (1.967 for a 98 percent confidence level) multiplied by the standard error from the sample proportion. This interval provides an estimate of the true population proportion with 98 percent confidence.
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What is the area of the following circle if the diameter is 2?
Answers
Answer:
Area of a circle=πr²
radius is half of the diameter
1/2×2=1
π1²
22/7×1
3.1415926535897932
~3.14
1106.666667 to the nearest whole number
Answers
Answer:
1106.666667 to the nearest whole number
1107
Answer:
1106.666667 rounds to 1107
Step-by-step explanation:
We look at the whole number, 6
Then look at the tenths place .6
Since 6 is greater than or equal to 5, we round the whole number up one place to 7
1106.666667 rounds to 1107
we have the number of rinks resurfaced by zzz zambonis is equal to the rate for zzz zambonis times the time. we want the time for 1\,\text{rink}1rink1, start text, r, i, n, k, end text , and we want that time to be less than or equal to 5\,\text{minutes}5minutes5, start text, m, i, n, u, t, e, s, end text. therefore: 1\,\text{rink}
Answers
we want that time to be less than equation or equal to 5 minutes start text, m, i, n, u, t, e, s, end text. therefore: 1\(5minutes\)
Rate for ZZZ Zambonis × Time ≤ 5 minutes
Time ≤ 5 minutes/Rate for ZZZ Zambonis
To solve for the time needed to resurface one rink, we first need to set up an equation. We know that the number of rinks resurfaced by ZZZ Zambonis is equal to the rate for ZZZ Zambonis multiplied by the time. We want the time for 1 rink to be less than or equal to 5 minutes, so we need to solve for time in the equation. We can do this by dividing both sides of the equation by the rate for ZZZ Zambonis, resulting in the equation Time ≤ 5 minutes/Rate for ZZZ Zambonis. This equation gives us the maximum time needed to resurface one rink.
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2 3/4 of 500grams in step by step calculator
Answers
Answer:
To calculate 2 3/4 of 500 grams, follow these steps:
1. Convert the mixed number to an improper fraction:
2 3/4 = (2 x 4 + 3)/4 = 11/4
2. Multiply the improper fraction by 500:
11/4 x 500 = (11 x 500)/4 = 2,750/4
3. Simplify the fraction by dividing the numerator and denominator by their greatest common factor, which is 2:
2,750/4 = (2 x 1,375)/(2 x 2) = 1,375/2
Therefore, 2 3/4 of 500 grams is equal to 1,375/2 grams or 687.5 grams.
Step-by-step explanation:
Triangle PQR has vertex coordinates at P(4, 0), Q(5, 4), R(5, 1). If the triangle is translated so that Q′(0, 4), determine the translation direction and number of units.
5 units down
5 units up
5 units to the right
5 units to the left
Answers
The translation direction is 5 units to the left.
To determine the translation direction and number of units, we can compare the original coordinates of point Q (Q(5, 4)) with the new coordinates of Q' after the translation (Q'(0, 4)).
By comparing the x-coordinates, we can see that the x-coordinate of Q' is smaller than the x-coordinate of Q, which means the triangle was translated to the left.
By comparing the y-coordinates, we can see that the y-coordinate of Q' is the same as the y-coordinate of Q, which means there was no vertical translation.
Therefore, the translation direction is 5 units to the left.
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Is the point (2,14) a solution of the equation y = 5x - 3?
Answers
Answer: No it is not