Let n be the last digit of your register number. Consider the initial value problem y" + 4y = 4un (t), y(0) = 0, y'(0) = 1.
a. Find the Laplace transform of the solution y(t).
b. Find the solution y(t) by inverting the transform.
Answers
To solve the initial value problem y" + 4y = 4u_n(t), where y(0) = 0 and y'(0) = 1, we will follow these steps:
a. Find the Laplace transform of the solution y(t).
The Laplace transform of the given differential equation can be obtained using the properties of the Laplace transform. Taking the Laplace transform of both sides, we get:
s^2Y(s) - sy(0) - y'(0) + 4Y(s) = 4U_n(s),
where Y(s) represents the Laplace transform of y(t) and U_n(s) is the Laplace transform of the unit step function u_n(t).
Since y(0) = 0 and y'(0) = 1, the equation becomes:
s^2Y(s) - s(0) - 1 + 4Y(s) = 4U_n(s),
s^2Y(s) + 4Y(s) - 1 = 4U_n(s).
Taking the inverse Laplace transform of both sides, we obtain the solution in the time domain:
y''(t) + 4y(t) = 4u_n(t).
b. Find the solution y(t) by inverting the transform.
To find the solution y(t) in the time domain, we need to solve the differential equation y''(t) + 4y(t) = 4u_n(t) with the initial conditions y(0) = 0 and y'(0) = 1.
The homogeneous solution to the differential equation is obtained by setting the right-hand side to zero:
y''(t) + 4y(t) = 0.
The characteristic equation is r^2 + 4 = 0, which has complex roots: r = ±2i.
The homogeneous solution is given by:
y_h(t) = c1cos(2t) + c2sin(2t),
where c1 and c2 are constants to be determined.
Next, we find the particular solution for the given right-hand side:
For t < n, u_n(t) = 0, and for t ≥ n, u_n(t) = 1.
For t < n, the particular solution is zero: y_p(t) = 0.
For t ≥ n, we need to find the particular solution satisfying y''(t) + 4y(t) = 4.
Since the right-hand side is a constant, we assume a constant particular solution: y_p(t) = A.
Plugging this into the differential equation, we get:
0 + 4A = 4,
A = 1.
Therefore, for t ≥ n, the particular solution is: y_p(t) = 1.
The general solution for t ≥ n is given by the sum of the homogeneous and particular solutions:
y(t) = y_h(t) + y_p(t)
y(t) = c1cos(2t) + c2sin(2t) + 1.
Using the initial conditions y(0) = 0 and y'(0) = 1, we can determine the values of c1 and c2:
y(0) = c1cos(0) + c2sin(0) + 1 = c1 + 1 = 0,
c1 = -1.
y'(t) = -2c1sin(2t) + 2c2cos(2t),
y'(0) = -2c1sin(0) + 2c2cos(0) = 2c2 = 1,
c2 = 1/2.
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Related Questions
Solve the inequality for w .
−9>6+5w
Answers
Hi :)
Let's use some Algebra skills to find w ^•^
————————————First off, let's subtract both sides by 6
\(\longrightarrow\darkblue\sf{-9-6 > 5w}\)
\(\longrightarrow\darkblue\sf{-15 < 5w}\)
Divide both sides by 5
\(\longrightarrow\sf{-3 < w}\)
Or
\(\longrightarrow\sf{w > -3}\)
\(\tt{Learn\:More;Work\:Harder}\)
:)
Jun Wei bought 15pencils at 2x cents each and 24 pens at 4y cents each. Find the total amount of money he paid for the pencils and pens
Answers
Answer:
124xy
Step-by-step explanation:
15 x 2x = 30x & 24 x 4y =96y 96y + 30x = 124 xy
Check the picture below, please answer thoroughly
Answers
Having calculated the Area of the various pieces of land and those of the respective individuals, we can conclude that
Hector owned a total of 810,000m² of land ; and Silvia go a larger share of the land.How is this so?Note that in each case, the land has been further divided in to two
a - Rectangular form
b - Right Triangle form
c - right Triangle form
d - Rectangular form
Thus,
Area of a from Ivan's land
= LB
600 x 400 = 240,000m²
Area of b from Ivan's land
= (bh)/2
= (600 * 500)/2
= 150,000m²
Thus the total area of Ivan's land =
240,000m² + 150,000m²
= 390 000 m²
Next lets do Silvias land
Area of c on slivia's land
= (bh)/2
= (600 * 500)/2
= 150,000m²
Area of d on Slivia's land
= 540 x 500
= 270,000m²
Thus the total area of Silvia's land
= 150,000m² + = 270,000m²
= 420,000m²
Thus, Hector owned a total of
= 390 000 m² + 420,000m²
= 810,000 m²
Since 420,000m² > 390 000 m²
we can conclude that
Silvia go the larger share of land.
See the attached.
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Based on the measurements given, DE is parallel to CB.
True
False
Answers
Answer:
True
Step-by-step explanation:
Measurements match up in proportion, indicating the same slope, meaning that the lines are parallel.
6*3 = 18
8*3 = 24
May I please have a Brainliest? I put a lot of thought and effort into my answers, so I would really appreciate it!
Question is in image answer question 1
Answers
you can check by plugging in a point
4 = -6(-2) -8
4 = 12-8
4 = 4
Answer:
y = -6x -8
Step-by-step explanation:
4 = -6(-2) -8
4 = 12-8
4 = 4
The explaination above shows how to verify your answer.
Hope this helps u!
In regression, what we want to establish is the exact numerical relationship between the two variables so that, for any given profit centre, we can try to forecast profit based on some causal value.
Select one:
True/False
Whereas time-series and causal models rely on quantitative data, qualitative models attempt to incorporate judgmental or subjective factors into the forecasting model.
Select one:
True/False
Answers
In regression, what we want to establish is the exact numerical relationship between the two variables so that, for any given profit centre, we can try to forecast profit based on some causal value. This statement is true.
Regression is a statistical tool that is utilized to establish the relationship between two variables. It examines whether there is a cause-and-effect connection between the two variables. It is commonly used in econometrics and finance to forecast and predict the future of a product or a business. The relationship between two variables is represented graphically on a scatter plot with regression analysis. Whereas time-series and causal models rely on quantitative data, qualitative models attempt to incorporate judgmental or subjective factors into the forecasting model. This statement is false. Qualitative methods, also known as judgmental techniques, rely on expert opinion and subjective information to make forecasts. Whereas time-series and causal models rely on quantitative data, qualitative models attempt to incorporate judgmental or subjective factors into the forecasting model.
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Obtain the general solution to the equation. +rtan 0 = 6 sec 0 tan 0 de The general solution is r(0)=, ignoring lost solutions, if any.
Answers
The general solution to the equation +rtan 0 = 6 sec 0 tan 0 de is r = 6/cos 0.
This means that r can take any value that satisfies this condition, as long as there are no lost solutions.
To obtain the general solution, we start by simplifying the equation using trigonometric identities. We know that sec 0 = 1/cos 0, and we can substitute this into the equation to get:
r tan 0 = 6/cos 0 tan 0
Dividing both sides by tan 0, we get:
r = 6/cos 0
This is the general solution to the equation, as r can take any value that satisfies this condition. However, it is important to note that there may be lost solutions, which occur when the simplification process involves dividing by a variable that may be equal to zero for certain values of 0. Therefore, it is important to check for such values of 0 that may result in lost solutions.
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With the information given, can you prove that this quadrilateral is a parallelogram?
Yes or No
Answers
Answer:
yes
Step-by-step explanation:
You want to know if the quadrilateral can be proven to be a parallelogram if opposite angles are congruent.
AnglesThe angles of a quadrilateral have a sum of 360°. If there are two pairs of congruent angles, the sum of angles of each pair must be 180°. If adjacent angles are supplementary, then opposite sides are parallel. A quadrilateral with opposite sides parallel is a parallelogram.
Yes, the figure can be proven to be a parallelogram.
adam is racing around a circular running track. the time he takes to run each lap is 5 seconds less than he took for the previous one. he completes the first lap in 1 minute and 58 seconds. how long (in seconds) does he take to run his seventh lap?
Answers
Time taken by Adam to complete his seventh lap of the circular track as per given data is equal to 1minute 28seconds.
As given in the question,
Time taken by Adam to complete hi first lap of circular track = 1 min 58 sec
First term t₁ = 1 minute 58seconds
Time taken by each successive lap is 5seconds less then the previous lap
Common difference 'd' = -5seconds
let time taken to complete the seventh lap be t₇
t₇ = t₁ + ( 7 - 1 ) d
⇒ t₇ = 1 minute 58 seconds + ( 6 ) (- 5seconds)
= 1 minute 58 seconds - 30seconds
= 1 minute 28 seconds
Therefore, the time taken by Adam to complete his seventh lap is equal to
1 minute 28 seconds.
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Can you please tell me what’s wrong in this plot?
I can’t see what.
Answers
Answer:
The Upper Quartile would not be 59. It would be 60.5.
Step-by-step explanation:
The Upper Quartile would not be 59. It would be 60.5.
Compare the y-intercepts for the two linear functions represented below.
X Y
-1 -1
0 1
1 3 y = 4x - 1
2 5
3 7
A.
The y-intercept of function A is same as the y-intercept of function B.
B.
The y-intercept of function A is greater than the y-intercept of function B.
C.
The y-intercept of function A is less than the y-intercept of function B.
Answers
B. The y-intercept of function A is greater than the y-intercept of function B.
What is a straight line?A straight line is an endless one-dimensional figure that has no width. It is a combination of endless points joined both sides of a point and has no curve.
here, we have,
from the given data, we get,
the 1st line's equation is, y=2x+1
i.e. y-intercept = 1
and,
the 2nd equation is,
1 3 y = 4x - 1
i.e. y-intercept = -1/13
=-0.77
as, 1> -0.77
Hence, B. The y-intercept of function A is greater than the y-intercept of function B.
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Check all that apply for the series [infinity] 12 n n=1 Diverges by the Divergence Test (nth term test). Convergent Geometric series. Divergent Geometric series. Divergent Harmonic series. Convergent Alternating Harmonic Series. Convergent p-series. Divergent p-series. Convergent by Comparison/Limit Comparison Test. Divergent by Comparison/Limit Comparision Test. Convergent by Alt. Series Test. Convergent by Ratio/Root Test. Divergent by Ratio/Root Test.
Answers
In general, the Divergence Test (nth term test) only allows us to determine whether a series diverges or not. It does not help us to determine the convergence of a series. Therefore, none of the other tests apply to this series.
The Divergence Test (nth term test) states that if the limit of the nth term of a series is not equal to zero, then the series diverges.
The series [infinity] 12 n n=1 is defined as follows:
[infinity] 12 n n=1 = 12¹ + 12² + 12³ + ...
The nth term of this series is given by:
aₙ = 12ⁿ As n → ∞, aₙ → ∞,
which means the limit of the nth term of the series does not exist.
Therefore, the series [infinity] 12 n n=1 diverges by the Divergence Test (nth term test).
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The estimate on an explanatory variable is not statistically significant at the 5%-level if the t-statistic is greater than 2.5. the p-value is less than 0.05. the 95% confidence interval does not contain zero. the 95% confidence interval contains zero.
Answers
The estimated value of the explanatory variable is not statistically significant at the 5%-level if a. the t-statistic is greater than 2.5.
The t-test is a hypothesis testing technique used to decide whether the estimated coefficient of a linear regression model is statistically significant or not. The t-value is used to measure the size of the difference between the estimated coefficient and the hypothesized value. When the t-value exceeds a critical value, such as 2.5, the estimated value of the explanatory variable is not significant at the 5%-level. It implies that the null hypothesis, which assumes that the explanatory variable's coefficient is zero, cannot be rejected.
The p-value is a measure of the probability of obtaining a more extreme value of the test statistic than the observed value if the null hypothesis is true. A small p-value implies that the null hypothesis should be rejected. When the p-value is less than 0.05, it means that the estimated value of the explanatory variable is significant at the 5%-level.
The confidence interval is a range of values that is expected to include the true value of the population parameter with a specified level of confidence. A 95% confidence interval means that the true value of the population parameter lies within the range 95% of the time. The 95% confidence interval does not contain zero implies that the estimated value of the explanatory variable is statistically significant at the 5%-level since it suggests that the population parameter is different from zero at the 5%-level. Conversely, if the 95% confidence interval contains zero, the estimated value of the explanatory variable is not statistically significant at the 5%-level since it means that the population parameter is not different from zero at the 5%-level. Therefore, it is essential to examine the t-statistic, p-value, and confidence interval while conducting hypothesis testing to make an informed decision.
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how do i Solve: X-1<3
Answers
Answer: x = 2 OR 3
Step-by-step explanation:
I haven't done this maths in a long time but when you look at this symbol < just imagine it's crocodile's mouth. The crocodile likes bigger meals (the bigger number which is 3). So, if you do 2 - 1 = 1 or 3 - 1 = 2, both 1 and 2 are smaller than 3.
Answer:
x<4
Step-by-step explanation:
x−1<3
Add 1 to both sides.
x<3+1
Add 3 and 1 to get 4.
x<4
Counting jails and prisons, approximately how many citizens are incarcerated? a. 1 million b. 2.3 million c. 3 million d. 4.3 million.
Answers
In September 2021, the approximate number of citizens incarcerated in jails and prisons is around 2.3 million.
It is important to note that this figure can vary over time due to changes in policies, criminal justice reforms, and other factors. The incarcerated population includes individuals who have been convicted of crimes and are serving their sentences, as well as those who are awaiting trial or have been sentenced but not yet transferred to a correctional facility.
These numbers can vary between different countries and jurisdictions. To obtain the most accurate and up-to-date information on current incarceration rates, it is advisable to refer to official sources such as the U.S. Bureau of Justice Statistics or relevant governmental organizations in your country.
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a data set has its first and third quartiles as 9 and 17 respectively. Which of the following data points would be considered an outlier for the data set
A. 27
B. 17
C. 3
D. 41
In which of these cases should the mean be used?
A. When the data is left-skewed
B. When the data is symmetric
C. When the data is right-skewed
D. When the data has extreme values
Answers
To determine if a data point is considered an outlier for a data set, we need to calculate the interquartile range (IQR) and use it to define the outlier boundaries. The IQR is the difference between the third quartile (Q3) and the first quartile (Q1). The correct option is (B).
We have that Q1 = 9 and Q3 = 17, we can calculate the IQR as follows:
IQR = Q3 - Q1 = 17 - 9 = 8
To identify outliers, we can use the following rule:
- Any data point that is less than Q1 - 1.5 * IQR or greater than Q3 + 1.5 * IQR is considered an outlier.
Using this rule, we can evaluate each data point:
A. 27: This data point is greater than Q3 + 1.5 * IQR = 17 + 1.5 * 8 = 29. It is considered an outlier.
B. 17: This data point is not an outlier because it is equal to the third quartile (Q3).
C. 3: This data point is less than Q1 - 1.5 * IQR = 9 - 1.5 * 8 = -3. It is considered an outlier.
D. 41: This data point is greater than Q3 + 1.5 * IQR = 17 + 1.5 * 8 = 29. It is considered an outlier.
Therefore, the outliers in the data set are A (27) and D (41).
As for when to use the mean, it is generally recommended to use the mean as a measure of central tendency when the data is symmetric and does not have extreme values.
Therefore, the correct option would be B. When the data is symmetric.
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what is
Which expression is equivalent to 9y – 3y?
3
3
3y
3y
6
6
6y please help
Answers
Answer:
9y-3y=6y is equivalent to 9y – 3y.
using these two points, (3, 6) and (4, 9); what is the slope?
Answers
EXPLANATION:
y2-y1
____= 3
x2-x1
evaluate tan 60/ cos 45
Answers
Answer:
0.60922709536
Step-by-step explanation:
not sure if this is what you are looking for????
√6
im putting yall on folks
Sylvia invested $500 in an account compounded annually with an interest rate of 8%. manuel invested $600 in an account with a compound interest rate of 7.25%. using the rule of 72, startfraction 72 over t endfraction, who will double their money first?
a. sylvia will double her money first, in approximately 9 years.
b. manuel will double his money first, in approximately 10 years.
c. manuel will double his money first, in approximately 9 years.
d. sylvia will double her money first, in approximately 10 years.
Answers
Using the rule of 72, startfraction 72 over t endfraction, Manuel will double his money first, in approximately 9 years. Option C is the answer.
The rule of 72The rule of 72 is a quick and simple way to estimate the number of years it takes for an investment to double given the
interest rate. It is calculated by dividing 72 by the interest rate.
For example, if an investment has an interest rate of 8%, it will take approximately 72 / 8 = 9 years to double the investment.
Applying this rule, we can estimate the time it will take for Sylvia's investment to double: 72 / 8 = 9 years.
And for Manuel's investment to double: 72 / 7.25 = 10 years.
Since Manuel's investment will double in approximately 9 years, he will double his money first.
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Math division help if can
Answers
Answer: A. 2 17 /21
Step-by-step explanation:you divide them.You change both mixed number to improper fraction.which is 236 /9 divide 28 /3.You do kcf which stand for KEEP CHANGE FLIP.YOU KEEP 236/9 and change Divide to mulipction sign and flip 28/3 to 3/28.You multiply236 /9 and 28/3 which gives u 708 /252.After that u simply it to 59 /21 .Lastly change back to mix number which is 2 17/21.
Hope it helps bye ;)
In 40 minutes; diane walks 2.5 miles and sue walks 1.5 miles. in miles per hour , how much far diane walking?
Answers
Diane will walk 1.5 miles more than Sue.
Firstly calculating the speed of Diane and Sue in miles per hour. As per the known fact, 1 hour has 60 minutes.
In 40 minutes Diane walks distance = 2.5 miles
So, converting speed in miles per hour = (2.5 ÷ 40)×60
Speed of Diane in miles per hour = 3.75 miles per hour
In 40 minutes Sue walks distance = 1.5 miles
So, converting speed in miles per hour = (1.5 ÷ 40)×60
Speed of Sue in miles per hour = 2.25 miles per hour
Difference between the distance covered = 3.75 - 2.25
Performing subtraction
Difference between the distance covered = 1.5 miles
Hence, Diane will walk 1.5 miles more than Sue.
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Esperanza is helping her aunt bag apples to sell at the market. They have 98 apples and they want to put 14 apples in each bag. How many full bags of apples will they have when they are done? a Make and explain a reasonable estimate for this problem. About how many bags will they fill? How do you know?
Answers
Answer:
They will have 7 full bags of apples.
Step-by-step explanation:
if you divide the 98 apples into groups of 14 you will have 7 groups (bags) of apples.
what is the median of 1,2,3,4,6
Answers
Answer:
Step-by-step explanation:
3
Consider this system of two linear equations.
5x − 2y = 16
-4x + 6y = -15
What is the y-coordinate of the solution to the system of equations?
Answers
SOME ONE PLEASE ACTUALLY HELP ME
Answers
Answer:
y = 658.3 ft
Step-by-step explanation:
x = 745.5 ft
y = 658.3 ft
tan 28 = 350/y
y = 658.3
Answered by GAUTHMATH
How do i solve this.
Answers
Let us set up some variables:
L: LengthW: WidthNow let us set up some equations based on the facts:
length of rectangle is 4 times its width --> L = 4Wperimeter of rectangle is 80 --> 2L + 2W = 80Equations:
L = 4W -- equation 1
2L + 2W = 80 -- equation 2
Plug the value of L from equation 1 into equation 2
2(4W) + 2W = 80
8W + 2W = 80
10W = 80
W = 8 -- equation 3
Plug the value of W from equation 3 into equation 1:
L = 4W = 4 * 8 = 32
Therefore the length is 32 and the width is 8.
Hope that helps!
1) solve 3x3 algebra equation5x−3y−4z=13
2y−4z=-22
−2z=-8
Answers
The solution to the given system of equations is:
x = 4, y = -3, z = 4.
To solve the given system of equations:
5x - 3y - 4z = 13
2y - 4z = -22
-2z = -8
We can start by solving equation (3) for z:
-2z = -8
Dividing both sides by -2:
z = (-8) / (-2)
z = 4
Now, substitute z = 4 into equations (2) and (1) to solve for y and x, respectively:
2y - 4z = -22
2y - 4(4) = -22
2y - 16 = -22
2y = -22 + 16
2y = -6
Dividing both sides by 2:
y = -6 / 2
y = -3
5x - 3y - 4z = 13
5x - 3(-3) - 4(4) = 13
5x + 9 - 16 = 13
5x - 7 = 13
5x = 13 + 7
5x = 20
Dividing both sides by 5:
x = 20 / 5
x = 4
Therefore, the solution to the given system of equations is:
x = 4, y = -3, z = 4.
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How old was Saul 11 years ago? (Represent answer in terms of x)
Answers
x is his present age in years
his age 11 years ago is x-11
Compound interest! Compound semiannually please
