Solve the differential equation y′′+2y′+y=e−tlnt by variation of parameters.
Answers
The general solution of the differential equation is the sum of the homogeneous and particular solutions:
\(y = y_h + y_p = c1 e^(-t) + c2 t e^(-t) + (t + 2 t ln t - 2 t + C) e^(-t)\)
where c1, c2, and C are constants to be determined by initial conditions.
We start by finding the general solution of the homogeneous differential equation y'' + 2y' + y = 0, which has the characteristic equation \(r^2 + 2r + 1 = 0.\)
Factoring the left-hand side, we get\((r + 1)^2 = 0\) , which has a repeated root r = -1.
Therefore, the general solution of the homogeneous equation is \(y_h = c1\) \(e^(-t) + c2 t e^(-t)\) , where c1 and c2 are constants to be determined by initial conditions.
To find a particular solution of the nonhomogeneous equation y'' + 2y' + \(y = e^(-t ln t)\), we assume a solution of the form \(y_p = u(t) e^(-t)\),
where u(t) is an unknown function to be determined.
We differentiate this expression to get\(y'_p = (u'(t) - u(t)) e^(-t) and y''_p = (u''(t) - 2u'(t) + u(t)) e^(-t).\)
Substituting these expressions into the differential equation, we get:
\((u''(t) - 2u'(t) + u(t)) e^(-t) + 2(u'(t) - u(t)) e^(-t) + u(t) e^(-t) = e^(-t ln t)\)
Simplifying, we get:
\(u''(t) = e^(t ln t)\)
Integrating once, we get:
\(u'(t)\) = ∫ \(e^(t ln t) dt\)
Using the substitution \(u = t ln t, du/dt = ln t + 1,\) and \(dt = du/(ln t + 1),\) we can write this integral as:
\(u'(t) =\) ∫ \(e^u / (ln t + 1) du\)
Using partial fractions, we can write:
\(e^u / (ln t + 1) = A + B ln t\)
where A and B are constants to be determined.
Multiplying both sides by ln t + 1 and setting ln t = -1, we get:
1 = A - B
Multiplying both sides by ln t and setting ln t = 0, we get:
\(e^u = A.\)
Therefore, \(A = e^u = 1\) and B = 2.
Substituting these values into the expression for u'(t), we get:
\(u'(t) = 1 + 2 ln t.\)
Integrating again, we get:
\(u(t) = t + 2 t ln t - 2 t + C\)
where C is a constant of integration.
Therefore, the particular solution of the differential equation is:
\(y_p = u(t) e^(-t) = (t + 2 t ln t - 2 t + C) e^(-t).\)
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Related Questions
The Diamond Rule says two factors, m and n, must multiply to get to the top number and add to get to the bottom number What are the possible values for m and n
-4 and -8
-2 and 16
4 and 8
-4 and 8
2 and 16
-2 and -16
Answers
Answer:
The answer for this should be A -4 and -8
Step-by-step explanation:
I'm also in k12
Answer
its 4 and 8 i took the quiz
Step-by-step explanation:
What the answer to this
Answers
Given:
The value of x = 2.
Solution:
F ( x ) = 2x⁴ - x³ + 5x - 9
=> 2 ( 2 )⁴ - ( 2 )³ + 5 × 2 - 9
=> 2 × 16 - 8 + 10 - 9
=> 32 - 8 + 10 - 9
=> 24 + 1
=> 25
A. 25
Please help!! I'll mark BRAINLIEST
Answers
Answer: C: 2(n+5)
Step-by-step explanation:
Answer:
2(n+5)Step-by-step explanation:
For finding we go from
Back to front that is,
First,
Sum of a number and 5 that is = n + 5
Then,
Twice of their sum that is = 2(n+5)
Hence,
Option C that is 2(n+5) (Ans)
A scale drawing shows a circular fountain with a diameter of 12 units. The actual fountain has a diameter of 60 feet. Select all the possible scales that could be used for the drawing.
Answers
Answer:i think its 70 if im wrong im very sorry
Step-by-step explanation:
Help me with is ixl please
Answers
Answer:
∠EGC
Step-by-step explanation:
Just like for the other problem, you're looking for an angle that's opposite the one you're given and is the same size as the one you're given (meaning they're congruent). If you use you hands to cover the parts of the line AD, you can see how ∠FGB and ∠EGC are opposite of each other and are congruent..
The prism below is made of cubes which measure of a centimeter on one side.
Which of the following represents the volume of the prism?
There are multiple answers.
3/2 cubic cm
1/216 cubic cm x 24
4/3 cubic cm
1/18 cubic cm x 24
1/9 cubic cm
(4 x 1/6cm) + (2 x 1/6cm) + (3 x 1/6cm)
(4 x 1/6cm) X (2 x 1/6cm) X (3 x 1/6cm)
NO LINKS NO FILES
Answers
The prism has an 8/27 cubic centimetre volume.
What is the volume of a cube?The following formula determines a cube's volume:
V = s³
where V is the volume of the cube and s is the length of one of its sides.
In other words, to find the volume of a cube, you simply need to raise the length of one of its sides to the third power (cube it).
The volume of a cube = s³
Given that the prism is composed of cubes with a one-third-centimeter side measurement.
Volume of a cube = (1/3)³ = 1/27
There are 8 cubes in the prism.
Therefore, the volume of the prism is;
=8(1/27)
= 8/27 cubic centimeter.
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The complete question is:
The prism below is made of cubes which measure of a centimeter on one side. What is the volume?
Note: Figure is not drawn to scale.
The prism below is made of cubes which measure 1/3 of a centimeter on one side. What is the volume?
Image is attached below.
50 points
answer this questions based on the graph :
-what does the point B represent?
-what is the unit rate of relationship shown on the graph
- how long does it take to drive 90 miles?
- how many miles can be driven in 6 hours?
Answers
Answer:
1) he will drive 180 miles in 4 hours
2) 45 mph
3) 2 hours
4) 270 miles
Step-by-step explanation:
Number 1:
The point is on (4,180) and since the x is hours and the y is miles. In 4 hours he can drive 180 miles.
Number 2:
For the unit rate we will use the same point (4,180) and put it the y over the x and simplify. \(\frac{y}{x}=\frac{180}{4}=\frac{45}{1}=45\) so in 1 hour he can drive 45 miles.
Number 3:
Now for this we should set up an equation \(y=45x\). So they told us if he drove for 90 miles that means we will substitute the y and place the 90 because as I said in number 1 y is miles.
\(y=45x\\90=45x\\x=2\)
So it'll take him 2 hours to drive 90 miles.
Number 4:
Now they are telling us how many miles can he drive if he drives for 6 hours. We can use the same equation for this and instead we will be substituting for x since x is hours.
\(y=45x\\y=45(6)\\y=270\)
So if he would to drive for 6 hours he can drive 270 miles.
a store keeper is paid 10% on all good sold. what commission will he receive if he sells goods worth gh¢ 1225.00?
Answers
The storekeeper will receive a commission of GH¢ 122.50 if he sells goods worth GH¢ 1225.00.
To calculate the commission that the storekeeper will receive, we need to multiply the value of the goods sold by the commission rate.
The commission rate is given as 10%, which can be expressed as 0.10 in decimal form.
Let's calculate the commission:
Commission = Value of goods sold * Commission rate
Commission = GH¢ 1225.00 * 0.10
Commission = GH¢ 122.50
Therefore, the storekeeper will receive a commission of GH¢ 122.50 if he sells goods worth GH¢ 1225.00.
The commission is calculated by taking 10% of the total value of goods sold. In this case, the storekeeper is entitled to a commission that amounts to 10% of the sales.
It's important to note that commission rates can vary depending on the agreement between the storekeeper and the employer. In this scenario, the given commission rate is fixed at 10%. If the commission rate were different, the calculation would be adjusted accordingly.
Commission is a form of incentive or compensation provided to the storekeeper based on the sales performance. It motivates the storekeeper to sell more and contributes to the overall profitability of the business. The specific commission structure and rates may vary across different industries and organizations.
In summary, the storekeeper will receive a commission of GH¢ 122.50 if he sells goods worth GH¢ 1225.00 at a commission rate of 10%.
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Find the coordinates of the midpoint of the segment whose endpoints are H(8,4) and K(6, 12).
Answers
Answer:
(7, 8)
Step-by-step explanation:
To find the mid point of two end point segments, you must first average out the x's of the coordinates; (8+6)/2
Next you do the same thing to the Y coordinate
(4+12)/2
Lastly you put them into a coordinate pair.
(7,8)
i need help answering with steps
-40-2(3m+1/2) =7m-2
Answers
Answer:
Step-by-step explanation:
-Start by distributing the 2, -2(3m+1/2).
-Getting -6m-1, your equation should look like -40-6m-1=7m-2
-Combine -40 and -1 to get -41, and then add the -6m to the other side.
-The equation should look like -41=13m-2. Add the 2 to the -41 to get -39.
-Now, we have -39=13m. divide both sides by 13 to get m=-3.
!! HELP ME PLEASE!!
*click the image*
What is the solution to the equation below?
Answers
Answer:
Solve for √3−2x.
√3−2x=4√x
To remove the radical on the left side of the equation, square both sides of the equation.
√3−2x²=(4√x)²
Simplify each side of the equation.
3−2x=16x
Solve for x.
=> x=16
Step-by-step explanation:
Answer:
x = \(\frac{1}{6}\)
Step-by-step explanation:
Given
\(\frac{\sqrt{3-2x} }{\sqrt{4x} }\) = 2 ( square both sides )
\(\frac{3-2x}{4x}\) = 4 ( multiply both sides by 4x )
3 - 2x = 16x ( add 2x to both sides )
3 = 18x ( divide both sides by 18 )
\(\frac{3}{18}\) = x , that is
x = \(\frac{1}{6}\)
What are your options in regards to the null hypothesis after you collect your data?
Answers
After you collect your data, you have several options in regards to the null hypothesis. First, you could fail to reject the null hypothesis.
The data did not provide enough evidence to support the alternative hypothesis, and you accept the null hypothesis as the most likely explanation. Second, you could reject the null hypothesis. This means that your data provided enough evidence to support the alternative hypothesis, and you reject the null hypothesis as the most likely explanation. To neither accept nor reject the null hypothesis. The less common option and occurs when your data does not provide enough evidence to reject the null hypothesis, but also does not provide enough evidence to accept it as the most likely explanation.
The null hypothesis after you collect your data include failing to reject it, rejecting it, or neither accepting nor rejecting it.
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Solve using substitution.
y = 6x + 2
y = 6x 10
Answers
Answer:
Step-by-step explanation:
1/27,20/9
The volumes of two similar solids are 512cm3 and 2197cm3. If the smaller solid has a surface are of 960cm2, find the surface area of the larger solid. Part 1: find the similarity ratio by taking the cube root of each volume. Show your work. Part 2: use your answer from part 1 to find the ratio of the surface areas. Show your work. Part 3: set up a proportion and solve to find the surface area of the larger solid.
Answers
Answer:
see below
Step-by-step explanation:
Part 1:
(512) ^ 1/3
-------------------
(2197) ^ 1/3
8
-----
13
The scale factor is 8:13
Part 2
The ratios of the areas is related by scale factor squared
8^2
-----
13^2
64
------
169
Part 3
64 960
------ = ----------------
169 SA larger
Using cross products
64 * SA = 169 * 960
64 SA = 162240
Divide each side by 64
64 SA/ 64 = 162240 / 64
SA = 2535
2535 cm^2
. You deposit $200 each month into an account earning 3% interest compounded monthly for 30 years. How much total interest will you earn in 30 years?
Answers
The total interest you will earn in 30 years is approximately $241.61.
To calculate the total interest earned in 30 years, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt) - P
Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
In this case, the principal amount is $200, the annual interest rate is 3% (or 0.03 as a decimal), the interest is compounded monthly (so n = 12), and the time period is 30 years (so t = 30).
Plugging in these values into the formula:
A = 200(1 + 0.03/12)^(12*30) - 200
Now we can simplify and calculate:
A = 200(1.0025)^(360) - 200
A = 200(2.208040033) - 200
A ≈ 441.6080066 - 200
A ≈ 241.6080066
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A compressive load of 80,000 lb is applied to a bar with
circular section0.75indiameter and a length of 10 in. if the
modulus of elasticity of the bar material is10,000 ksi and the
Poisson’s ratio i
Answers
The decrease in diameter of the bar due to the applied load is -0.005434905d and the final diameter of the bar is 1.005434905d.
A compressive load of 80,000 lb is applied to a bar with a circular section of 0.75 in diameter and a length of 10 in.
if the modulus of elasticity of the bar material is 10,000 ksi and the Poisson's ratio is 0.3.
We have to determine the decrease in diameter of the bar due to the applied load.
Let d be the initial diameter of the bar and ∆d be the decrease in diameter of the bar due to the applied load, then the final diameter of the bar is d - ∆d.
Length of the bar, L = 10 in
Cross-sectional area of the bar, A = πd²/4 = π(0.75)²/4 = 0.4418 in²
Stress produced by the applied load,σ = P/A
= 80,000/0.4418
= 181163.5 psi
Young's modulus of elasticity, E = 10,000 ksi
Poisson's ratio, ν = 0.3
The longitudinal strain produced in the bar, ɛ = σ/E
= 181163.5/10,000,000
= 0.01811635
The lateral strain produced in the bar, υ = νɛ
= 0.3 × 0.01811635
= 0.005434905'
The decrease in diameter of the bar due to the applied load, ∆d/d = -υ
= -0.005434905∆d
= -0.005434905d
The final diameter of the bar,
d - ∆d = d + 0.005434905d
= 1.005434905d
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The Medina's went out to eat at Chili's. If there bill was 65.20 and they gave 1p
their server a 15% tip, how much did they leave for a tip? *
Answers
Answer:
The Medinas left $9.78 for a tip
Step-by-step explanation:
Percentages
The Medinas went out to eat at Chilli's and the bill was $65.20. They tipped 15% to their server.
This percentage is calculated over the amount of the $65.20 bill, thus the tip was:
15% of 65.20 = 15/100 * $65.20 = $9.78.
The Medinas left $9.78 for a tip
Write down two factors of 24 that are primenumber
Answers
the prime factors of 24 are 2 and 3, which combine to give the unique prime factorization of 24 as 2^3 × 3.
There are no factors of 24 that are prime numbers. A factor of a number is a whole number that divides that number without leaving a remainder. Prime numbers, on the other hand, are numbers that are divisible only by 1 and themselves, and cannot be expressed as the product of any other numbers.
The prime factors of 24 are 2, 2, and 3. We can factorize 24 as 2 × 2 × 2 × 3 or 2^3 × 3. Here, 2 and 3 are both prime numbers, but they are not factors of 24 in isolation. They are only prime factors of 24 when combined in the manner shown.
This fact highlights an important concept in number theory: the uniqueness of prime factorization. Every composite number can be expressed as a unique product of prime numbers. This fundamental theorem of arithmetic is crucial in many areas of mathematics, including cryptography, where it is used to secure communications and protect sensitive information.
In summary, there are no factors of 24 that are prime numbers. However, the prime factors of 24 are 2 and 3, which combine to give the unique prime factorization of 24 as 2^3 × 3.
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Create a system where the solution is (3,-4)
4x+1 y=8
(?)x+(?)y=(?)
(3,-4)
Answers
The system of equations that satisfies the solution (3, -4) is:
4x + 1y = 8
2x - 3y = -17
How to Create a system where the solution is (3,-4)To create a system of equations where the solution is (3, -4), we can assign arbitrary values to the coefficients of the equations. Let's use the following values:
Equation 1: 4x + 1y = 8
Equation 2: 2x - 3y = -17
By plugging in the values (3, -4) into these equations, we can find the missing coefficients:
Equation 1: 4(3) + 1(-4) = 12 - 4 = 8
Equation 2: 2(3) - 3(-4) = 6 + 12 = 18 - 17 = -17
Therefore, the system of equations that satisfies the solution (3, -4) is:
4x + 1y = 8
2x - 3y = -17
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consider an undirected graph that has 100 vertices. for any pair of vertices, only 1 edge can connect them. in other words, you cannot have 2 edges connecting vertex a directly to vertex b. also assume that there are no self-loops, i.e. an edge that goes from vertex a to vertex a. what is the maximum number of edges that can be in this specific graph?
Answers
So the maximum number of edges in an undirected graph with 100 vertices is 4950.
In an undirected graph, each edge connects two vertices, and as such, it is counted twice, once for each vertex it connects. Therefore, the total number of edges in the graph is the sum of the degrees of all vertices, divided by 2.
In a complete graph with n vertices, every vertex is connected to every other vertex, except itself, and so the degree of each vertex is n-1 (it is connected to n-1 other vertices). Therefore, the total number of edges in the graph is:
n * (n-1) / 2
Substituting n = 100, we get:
100 * 99 / 2 = 4950
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The table compares the shoes worn by customers in an ice cream shop and the ice cream flavor that they picked. Can ice cream flavor be represented as a function of the type of shoe? Shoe type Ice cream flavor Boots Strawberry Loafers Vanilla Sandals Chocolate High-heels Banana Flip-flops Pineapple Moccasins Chocolate Sandals Banana Loafers Vanilla Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answers
the answer is no.
bajaabjshsiwhdd
Answer:
No
Step-by-step explanation:
what is 4 + 1
cause i'm stuck on this question hehe
Answers
Explernation:
Yee Welcome.
Have a great day!
Answer:
5
Step-by-step explanation:
4+1=5...if you add the number 4 and 1 it gives 5 lol
Jameela spins the following spinner 36 times.
A spinner with 4 equal sections labeled 1 through 4.
How many times should she expect it to land on 3?
Enter your answer as a number, like this: 42
PLEASE HELP ME IM NOT GOOD AT MATH
Answers
Using the binomial distribution, it is found that she should expect it to land on 3 9 times.
What is the binomial probability distribution?It is the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.
The expected value of the binomial distribution is:
E(X) = np
In this problem:
The spinner is equally as likely to land on any of the four sections, hence p = 1/4 = 0.25.There will be 36 trials, hence n = 36.Then, the expected value is given by:
E(X) = np = 36 x 0.25 = 9.
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factorise x²-49
please solve this answer
Answers
Answer:
(x+7)(x-7)
Step-by-step explanation:
x²-49
=x²-7²
=(x+7)(x-7)
Answer:
(x+7) (x-7)
Step-by-step explanation:
\(\sqrt x^{2}\)-49
\(\sqrt x^{2}\) - \(\sqrt{49}\) = (x+7) (x-7)
Simplify the expression: 8(3y + 5) - 5
24y + 35
24y - 45
24y
24y + 80
Answers
Answer:
24y + 35
Step-by-step explanation:
Hey there!
8(3y + 5) - 5
= 8(3y) + 8(5) - 5
= 24y + 40 - 5
= 24y + 35
Therefore, the answer is:
Option A. 24y + 35
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
You play a game and win 136 out of 270 times. Find a 95% confidence interval for the probability of winning the game.
Answers
To calculate the 95% confidence interval for the probability of winning the game, we can use the following formula:
Lower bound: \(\hat{p}-z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
Upper bound: \(\hat{p}+z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
where $\hat{p}$ is the sample proportion, n is the sample size, and \(z_{\alpha/2}\) is the z-score corresponding to the required level of confidence. The samples we received
\(\hat{p}=\frac{136}{270}\ approx0.5037\). The sample size is n=270. To find \(z_{\alpha/2}\), we can use a z-table or calculator. For a 95% confidence interval,
\(\alpha\)=1-0.95
=0.05,
so \(\alpha/2\)=0.025.
If we find the z-score of 0.025 in the z-table or using a calculator, we get \(z_{0.025}\approx1.96\). Converting the value to the formula, we assume:
Lower bound: \(0.5037-1.96\sqrt{\frac{0.5037(1-0.5037)}{270}}\approx0.4484\)
Upper bound: \(0.5037+1.96\sqrt{\frac{0.5037(1-0.5037)}{270}}\approx0.5589\)
Therefore, the 95% confidence interval for the probability of winning the game is (0.4484, 0.5589).
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What type of calculator is used in algebra 2?
Answers
In Algebra 2, students typically use a scientific calculator.
A scientific calculator has advanced functions beyond basic arithmetic operations.
Some of the functions commonly used in Algebra 2 include logarithms, exponents, trigonometric functions, and statistical calculations. Therefore, a scientific calculator with these functions would be most useful for Algebra 2.
Examples of calculators that are often used in Algebra 2 include the TI-84 Plus , the TI-Nspire CX, the Casio fx-9750GII, and the HP Prime Graphing Calculator.
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In simplest form, what is the quotient of 1/6 divided by 2/9?
Answers
step by step: The numbers in 1/6 divided by 2/9 are labeled below:
1 = dividend numerator
6 = dividend denominator
2 = divisor numerator
9 = divisor denominator
To make it a fraction form answer, you multiply the dividend numerator by the divisor denominator to make a new numerator.
Furthermore, you multiply the dividend denominator by the divisor numerator to make a new denominator:
1 x 9
6 x 2
=
9
12
Thus, the answer to 1/6 divided by 2/9 in fraction form is:
9
12
To make the answer to 1/6 divided by 2/9 in decimal form, you simply divide the numerator by the denominator from the fraction answer above:
9/12 = 0.75
The answer is rounded to the nearest four decimal points if necessary.
9/12 can be simplified to 3/4.
14% out of 100% equals how many people out of 10?
Answers
Answer:
1.4 people out of 10
Step-by-step explanation:
Just divide it by 10
Answer:
only 4 people
Match the following terms to their definition Feasible region [Choose ] Binding constraint [ Choose ] Sensitivity analysis [ Choose ] Constraint [Choose ] < Decision variables [Choose ] Objective function [ Choose ] Shadow price [Choose ]
Answers
Feasible region - the set of all possible solutions that satisfy all constraints.Binding constraint - a constraint that is met exactly at the optimal solution.Sensitivity analysis - the process of analyzing how changes in the parameters of a linear programming problem affect the optimal solution
1. Feasible region: A set of values for decision variables that satisfy all constraints in an optimization problem.
2. Binding constraint: A constraint that holds as an equality at the optimal solution, affecting the optimal value of the objective function.
3. Sensitivity analysis: A technique used to determine how different values of an independent variable affect a dependent variable under given constraints.
4. Constraint: A condition or limitation imposed on decision variables in an optimization problem.
5. Decision variables: The variables that represent the choices available to the decision-maker in an optimization problem.
6. Objective function: The mathematical expression representing the goal of an optimization problem, which is to be minimized or maximized.
7. Shadow price: The change in the objective function value due to a one-unit increase in the availability of a limited resource, holding other factors constant.
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