Find the solution of the initial value problem Y" – 2y – 3y = 15tet, y(0) = 2, y'(0) = 0. = = - 2 NOTE: Enter an exact answer. y(t) =

Answers

Answer 1

The solution of the initial value problem is:$Y(t) = e^(3t) + e^(-t) - 3tet$

To solve the initial value problem Y" – 2y – 3y = 15tet, y(0) = 2, y'(0) = 0, we can use the method of undetermined coefficients.

First, we find the general solution of the homogeneous equation Y" – 2y – 3y = 0.

The characteristic equation is:

$r^2 - 2r - 3 = 0$

Factoring the quadratic equation, we have:

(r - 3)(r + 1) = 0

This gives us two distinct roots: r = 3 and r = -1.

Therefore, the general solution of the homogeneous equation is:

$Yh(t) = C1e^(3t) + C2e^(-t)$

To find a particular solution Yp(t) for the non-homogeneous equation, we assume a solution of the form Yp(t) = Atet, where A is a constant to be determined.

Taking the first and second derivatives of Yp(t), we have:

$Yp'(t) = Ate^t + Aet$

$Yp"(t) = Ate^t + 2Aet$

Substituting these derivatives into the non-homogeneous equation, we get:

$(Ate^t + 2Aet) - 2(Atet) - 3(Atet) = 15tet$

Simplifying the equation, we have:

$Ate^t + 2Aet - 2Ate^t - 3Ate^t = 15tet$

Combining like terms, we get:

$(-4A + 2A - 3A)te^t = 15tet$

Simplifying further, we have:

$-5Ate^t = 15tet$

Cancelling out the common terms, we get:

-5A = 15

Solving for A, we find:

A = -3

Now, we have the particular solution Yp(t) = -3tet.

The general solution of the non-homogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution:

Y(t) = Yh(t) + Yp(t)

$Y(t) = C1e^(3t) + C2e^(-t) - 3tet$

Using the initial conditions y(0) = 2 and y'(0) = 0, we can solve for the values of C1 and C2.

When t = 0:

$Y(0) = C1e^(3(0)) + C2e^(-0) - 3(0)e^(0)$

2 = C1 + C2

Taking the derivative of Y(t) with respect to t and evaluating it at t = 0:

$Y'(t) = 3C1e^(3t) - C2e^(-t) - 3te^(3t)Y'(0) = 3C1e^(3(0)) - C2e^(-0) - 3(0)e^(3(0))$

0 = 3C1 - C2

Solving these equations simultaneously, we find C1 = 1 and C2 = 1.

Therefore, the solution of the initial value problem is:

$Y(t) = e^(3t) + e^(-t) - 3tet$

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Related Questions

how to show - 2by3 and 1by2 on the number line

Answers

Answer: Hope it help

Step-by-step explanation:

how to show - 2by3 and 1by2 on the number line

the completion times for a certain marathon race was 2.4 hours with a standard deviation of 0.5 hours. what can you determine about these data by using chebyshev's inequality with latex: k=2?

Answers

The answer is $P(1.4 \leq X \leq 3.4) \geq 1-\frac{1}{k^2}=1-\frac{1}{4}=0.75$, by using the Chebyshev's inequality with k=2.

Chebyshev's inequality applied to data sets of any type; that is, there is no need that the distribution of data is to be symmetric. And as per this inequality, the proportion of $(1-\frac{1}{k^{2} } )$ data values will fall within the k number of standard deviation from the value of mean.

Using Chebyshev's inequality with k=2, we can determine that at least 75% of the marathon runners completed the race within 3.4 hours (2.4 + 2*0.5). It means at least 75% of the marathon runners completed the race in less than 3.4 hours. Mathematically, this can be represented as:

$P(1.4 \leq X \leq 3.4) \geq 1-\frac{1}{k^2}=1-\frac{1}{4}=0.75$

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Jeremy and his classmates conduct an experiment with a set of weights and a spring. They record their results in the table.
Use the table to answer Items 21-24.
2
Student
Mass (lb)
Spring Stretch (in.)
Jeremy
5
7.5
Adele
I
12
Roberto
14
21
Shanice
21
36
Guillaume
28
42
21. Make a graph of the data.

Answers

Here's link to the answer:

tinyurl.com/wpazsebu

solve for w: 7w < -56

Answers

Lets solve

$\\ \sf\longmapsto 7w<-56$

Take 7 to right side

$\\ \sf\longmapsto w<dfrac{-56}{7}$

Simplify

$\\ \sf\longmapsto w<-8$

Answer:

$\boxed {\boxed {\sf w < -8}}$

Step-by-step explanation:

We are given an inequality and asked to solve for the variable w.

$7w < -56$

In order to solve for w, we must isolate the variable. It is being multiplied by 7. The inverse of multiplication is division, so we divide both sides of the inequality by 7.

$\frac {7w}{7} < \frac {-56}{7}$

$w < \frac{-56}{7}$

$w < -8$

w is less than -8 or w < -8.

Craig has 60 feet of fencing with which to make a rectangular garden area. One side of the rectangular garden will be the side of the Craig's house. Find the length and width for a maximum garden area

Answers

The length and width for a maximum garden area is 15 feet by 30 feet.

What is an equation?

An equation is an expression that shows the relationship between two or more numbers and variables. An independent variable is a variable that does not depend on other variables while a dependent variable is a variable that depends on other variables.

Let x represent the side parallel to Craig's house and y represent the other side. One side of the rectangular garden will be the side of the Craig's house. Hence:

2x + y = 60

y = 60 - 2x (1)

Area (A) = xy = x(60 - 2x)

A = 60x - 2x²

The maximum area is at A' = 0, hence:

A' = 60 - 4x

60 - 4x = 0

x = 15

y = 60 - 2(15) = 30

The length and width for a maximum garden area is 15 feet by 30 feet.

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Two cards are drawn from a deck of 52 playing cards. The first card is replaced into the deck before the 2nd card is drawn. What is the probability drawing Ace of Spades and any King?

Answers

Answer:

1/676

Step-by-step explanation:

There are 52 cards

P(Ace of spades) = Number of aces of spades / total = 1/ 52

The replace the card

There are 52 cards

There are 4 kings

P(king) = number of kings / total = 4/52 = 1/13

P( ace of spades , replace, any king) = 1/52 * 1/13 = 1 /676

What is the solution to the division problem below? (You can use long division or synthetic division.) (4x2 + 5x - 6) (x + 2)

Answers

Answer:

The other factor is 4x-3

Step-by-step explanation:

We want to divide 4x^2 + 5x -6 by x + 2

Unfortunately, we cannot say if x + 2 is a factor or not.

To know if it’s,

we set x + 2 = 0 and this means x = -2

We substitute this into the larger polynomial.

This gives;

4(-2)^2 + 5(-2) -6 = 16 -10-6 = 0

What this actually means is that it is a factor of it since it leaves no remainder.

The issue now is finding this other factor.

So let’s use the long division

(x + 2). _| 4x ^2 + 5x -6

Kindly note that 4x^2/x = 4x, we now multiply this by (x+2) so we get 4x(x+2) = 4x^2 + 8x which we will subtract from 4x^2 + 5x -6

_ 4x_

(x+2). ___|4x^2 + 5x -6

-(4x^2 + 8x)

-3x -6

next is -3x/x = -3

_-4x-3_

(x+2). _| 4x^2 + 5x-6

-(4x^2 +8x)

__________

-3x-6

-(-3x-6)

_________

So the other factor is 4x-3

given f(x)=-3x+3solve for x when f(x)=6

Answers

Answer:

x = -1

Step-by-step explanation:

Given: $f(x) = - 3x + 3$

Solve for x, When: $f(x) = 6$

Step by step:

$ - 3x + 3 = 6$

$ - 3x = 6 - 3$

$ - 3x = 3$

$x = 3 \div - 3$

$\boxed{\green{x = - 1}}$

what is the circumference and area of a circle with a radius of 4 meters

Answers

Answer:

Area = 50.27 m

Circumference = 25.13 m

Detailed Explanation:

Radius (r) = 4 m

Area of a Circle

= πr^2

= π * r^2 = π * r * r

= 3.14 * 4^2 = 3.14 * 16

=> 50.27 m

Circumference of the Circle

= 2πr

= 2 * π * r

= 2 * 3.14 * 4

=> 25.13 m

2.81551 nearest hundredth

Answers

2.82 is the nearest hundredth.

Hope this helps.

The answer is 2.82 because thousandths place is above 5, therefore the number before it rounds up

The table shows the results of a survey about students' preferred subjects in middle
school. What is the probability that a randomly selected student from the survey is a
7th grader that prefers English? Write your answer as a fraction and NOT as a
decimal.

Answers

Answer:

19/149

Step-by-step explanation:

38/298 = 19/149

7th graders who prefer English out of all students

What would be the opportunity cost of spending $90,000 on advertising but only producing 12,000 units? Potential sales (before advertising) of 12,000 units, Price of $16, Fixed costs of $48,000, Variable costs $8, Advertising $90,000 Assume advertising multiplier is (30,000+ advertising)/30,000
$76,800
$576,000
$192,000
−$191,936
$768,000

Answers

The opportunity cost of spending $90,000 on advertising but only producing 12,000 units can be calculated by comparing the benefits of the advertising investment to the potential alternative uses of that money.

First, let's calculate the total cost of producing 12,000 units. Fixed costs amount to $48,000, and variable costs are $8 per unit, resulting in a total cost of $48,000 + ($8 × 12,000) = $144,000.

Next, we need to calculate the potential sales revenue without advertising. With a price of $16 per unit, the potential sales revenue would be $16 × 12,000 = $192,000.

Now, let's calculate the potential sales revenue after advertising. The advertising multiplier is given as (30,000 + advertising) / 30,000. In this case, the multiplier would be (30,000 + 90,000) / 30,000 = 4.

Therefore, the potential sales revenue after advertising would be $192,000 × 4 = $768,000.

The opportunity cost is the difference between the potential sales revenue after advertising ($768,000) and the potential sales revenue without advertising ($192,000), which is $768,000 - $192,000 = $576,000.

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8.4g of sugar is needed for every cake made. How much sugar is needed for 8 cakes?

Answers

Answer:

67.2g of sugar will be needed to make 8 cake

Step-by-step explanation:

8.4g x 8 cake

= 67.2g sugars

find the lemgth of arc PQ

Answers

The arc length is D) 3.14 meters.

What is arc length?

In mathematics, an arc is a portion of a curve that can be thought of as a segment of the curve. An arc is a connected set of points on a curve, usually a portion of a circle.

It is defined by two endpoints and all the points along the curve between them. The length of an arc is the distance along the curve between its two endpoints.

The distance along the curved line that forms the arc (a section of a circle) is measured using the arc length formula.

$A_L=\frac{\theta}{360\textdegree}2\pi r$

Here the give circle Radius PR = 3m and θ=60°.

Now using arc length formula then,

=> Arc length $A_L=\frac{\theta}{360\textdegree}2\pi r$

=> $A_L=\frac{60}{360} \times2\times3.14\times3$

=> $A_L=\frac{1}{6}\times2\times3.14\times3$

=> $A_L$ = 3.14 m.

Hence the arc length is D) 3.14 meters.

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work out 3/8 of 720

andwer

Answers

Answer:

270

Step-by-step explanation:

$\frac{3}{8}$ of 720

= $\frac{3}{8}$ × 720

= 3 × $\frac{720}{8}$

= 3 × 90

= 270

(3/8)*720
0.375*720
=270

Marissa is walking along the beach and finds an unknown organism on the shore it has radical symmetry a nonporous surface and tentacles in which phylum can the organism be classified a.chordata b.Cnidaria c.echinodermata d.porifera

Answers

this is not math ? it’s physics or chem or bio or something

2) Triangle ABC has three sides that all measure 7 feet. What is the measure of Angle B? 3)

Answers

Answer:

Step-by-step explanation:

Answer:

60°

Step-by-step explanation:

PLEASE HELP!!!!!!!!!!!!!!!

Answers

Answer:

27.6%

Step-by-step explanation:

See the attached worksheet. Find the total number of marbles (29) and then take the green marbles and divide by the total and multiply by 100%. (8/29)*(100%) = 27.6% to the nearest tenth.

15 Find the area of the
shaded region.

Answers

Answer: 100in

Step-by-step explanation:

20x10=200

Volume of a Triangle is BasexHeight divided by 2

SoArea for triangle is 20x10/2 which is 100

then do 200-100 which will giveyou the answer

100!

100 inches

10 x 10=100 divided by 2=50

50x2=100

(a) Explain why a gamma random variable with parameters (n, λ) has an approximately normal distribution when n is large.
(b) Then use the result in part (a) to solve Problem 9.20, page 395.
(d) What does the central limit theorem say with continuity correction? (e) Find the exact probability. steps, find the probability that the walk is within 500 steps from the origin calculations, explain why X ︽.Norm(a/λ, a/λ2). 9.18 Consider a random walk as described in Example 9.13. After one million 9.19 Let X ~ Gamma(a,A), where a is a large integer. Without doing any 9.20 Show that lim Hint: Consider an independent sum of n Exponential() random variables and apply the central limit theorem. 9.21 A random variable Y is said to have a lognormal distribution if log Y has a normal distribution. Equivalently, we can write Y -eX, where X has a normal distribution. (a) If X1, X2,... is an independent sequence of uniform (0,1) variables, show that the product Y =「L-i X, has an approximate lognormal distribution. Show that the mean and variance of log Y are, respectively, -n and n (b) If Y = ex, with X ~ Norm(μ, σ2), it can be shown that

Answers

the gamma distribution becomes approximately normal due to the Central Limit Theorem when n is large.X ︽.Norm(a/λ, a/λ²) since it is an approximately normal distribution with mean a/λ and variance a/λ².

(a) Gamma random variables are sums of random variables, and as n gets large, the Central Limit Theorem applies. When n is large, the gamma random variable with parameters (n, λ) approaches a normal distribution, as the sum of independent and identically distributed Exponential(λ) random variables is distributed roughly as a normal distribution with mean n/λ and variance n/λ². In other words, the gamma distribution becomes approximately normal due to the Central Limit Theorem when n is large.

(b) The problem asks to show that:lim (1 + x/n)-n = e⁻x.The expression (1 + x/n)⁻ⁿ can be written as [(1 + x/n)¹/n]ⁿ. Now letting n → ∞ in this equation and replacing x with aλ yields the desired result from part (a):lim (1 + x/n)ⁿ

= lim [(1 + aλ/n)¹/n]ⁿ

= e⁻aλ(d)

The central limit theorem with continuity correction can be expressed as:P(Z ≤ z) ≈ Φ(z + 0.5/n)if X ~ B(n,p), where Φ is the standard normal distribution and Z is the standard normal variable.

This continuity correction adjusts for the error made by approximating a discrete distribution with a continuous one.(e) The exact probability that the walk is within 500 steps from the origin can be calculated by using the normal distribution. Specifically, we have that:

P(|X - a/λ| < 500)

= P(-500 < X - a/λ < 500)

= P(-500 + a/λ < X < 500 + a/λ)

= Φ((500 + a/λ - μ)/(σ/√n)) - Φ((-500 + a/λ - μ)/(σ/√n)),

where X ~ N(μ, σ²), and in this case, μ = a/λ and σ² = a/λ².

Therefore, X ︽.Norm(a/λ, a/λ²) since it is an approximately normal distribution with mean a/λ and variance a/λ².

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Can y’all please give me the answers I would truly appreciate it

Answers

Step-by-step explanation:

a.) So he has 13 coins and those coins are nickels and quarters. If we let n be nickels and q be quarters

n + q = 13 because the total number of quarters plus nickels is 13

The coins have a total value of $2.05

Nickels are 0.05 cents and quarters are 0.25 cents, which means

0.05n + 0.25q = $2.05

Making your system of equations

$n + q = 13 \\ 0.05n + 0.25q = 2.05$

Answer:

a.) So he has 13 coins and those coins are nickels and quarters. If we let n be nickels and q be quarters

Step-by-step explanation:

Betty has 42 butterfly stickers, as shown below.

She puts an equal number of stickers on each of 6 pages in her sticker book.

How many stickers does Betty put on each page in her sticker book?

Answers

Answer:

Step-by-step explanation:

42/6=7

Need help quick new topic need answers please

Answers

Answer:

circumference=2xpixradius

Ex: for #1, the radius is 6.4, so, to find the circumference you will do this:

2 times 3.14159265359 times 6.4= circumference.

Step-by-step explanation:

I will get reported if I give all the answers so I tried to help by saying how you can find the circumference answers to the questions.

c.) In a class of 28 sophomores and juniors, 16 are
juniors. What percent of the class are sophomores?

Answers

The percent of the sophomores in the class of 28 sophomores and juniors is 42.86%.

According to the question,

We have the following information:

Total number in a class including sophomores and juniors = 28

Number of juniors in the class = 16

Now, we have the number of sophomores in this class:

Total number in the class- Number of juniors in the class

28-16

Now, we can easily find the percent of sophomores in the class by following the given steps:

Number of sophomores*100/Number of sophomores and juniors

1200/28

42.86%

Hence, the percent of the sophomores in the class of 28 sophomores and juniors is 42.86%.

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Erica brought $30.25 to the state fair. she bought a burger, a souvenir, and a pass. the burger was 1 6 as much as the souvenir, and the souvenir cost 3 4 the cost of the pass. erica had $4.00 left over after buying these items. what was the cost of each item?

Answers

Let's break down the given information and solve the problem step by step:

Let's assume:

The cost of the burger is 'b' dollars.

The cost of the souvenir is 's' dollars.

The cost of the pass is 'p' dollars.

According to the given information:

The burger was 1/6 as much as the souvenir. This can be expressed as: b = (1/6) * s.

The souvenir cost 3/4 the cost of the pass. This can be expressed as: s = (3/4) * p.

Erica had $4.00 left over after buying these items. This can be expressed as: b + s + p + 4 = 30.25.

Now, let's solve these equations to find the cost of each item.

Substituting the value of 's' from equation 2 into equation 1:

b = (1/6) * ((3/4) * p)

b = (3/24) * p

b = (1/8) * p

Substituting the values of 'b' and 's' into equation 3:

(1/8) * p + ((3/4) * p) + p + 4 = 30.25

Multiply through by 8 to eliminate the fraction:

p + 6p + 8p + 32 = 242

15p + 32 = 242

15p = 210

p = 210 / 15

p = 14

Substituting the value of 'p' into equation 2 to find 's':

s = (3/4) * 14

s = 10.50

Substituting the value of 'p' into equation 1 to find 'b':

b = (1/8) * 14

b = 1.75

Therefore, the cost of the burger is $1.75, the cost of the souvenir is $10.50, and the cost of the pass is $14.0

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Might give you Brainliest is you step by step explain how much you get the answer

Answers

Answer:

Step-by-step explanation:

There were 160 votes in total, and we need to figure out how much is 3/8 of it. Therefore, we need to find (3/8)*(160) (of is another word for multiply). The easiest method is to find 1/8 of 160 which is 20, and then multiply it by three to get 60, which is your answer!

I hope this helped, please give brainliest if this is correct! Thank you!

In the diagram, M is the midpoint of AB.

Support the following argument for AM = MB by
writing the reason that validates each statement.

A. Definition of midpoint

B. Definition of congruent segments

C. Given information

We know that M is the midpoint of AB.

This means that AM = MB.

Therefore, AM=~MB.