Can somebody help me please no links

The series ∑[(k² + 2)/(k³ - 7)] diverges using the limit comparison test and the series ∑(1/k) is a well-known harmonic series.

To determine the convergence of the series ∑[(k² + 2)/(k³ - 7)], we can use the limit comparison test. Let's compare it with the series ∑(1/k).

First, we need to find the limit of the ratio of the terms of the two series as k approaches infinity:

lim(k→∞) [(k² + 2)/(k³ - 7)] / (1/k)

Simplifying the expression, we get:

lim(k→∞) [(k² + 2)/(k³ - 7)] × (k/1)

Taking the limit as k approaches infinity, we have:

lim(k→∞) [(k² + 2)/(k³ - 7)] × (k/1) = 1

Since the limit is a finite positive value (1), we can conclude that the given series ∑[(k² + 2)/(k³ - 7)] and the series ∑(1/k) have the same convergence behavior.

The series ∑(1/k) is a well-known harmonic series, which diverges. Therefore, by the limit comparison test, the given series ∑[(k² + 2)/(k³ - 7)] also diverges.

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The question is -

Use the limit comparison test to determine whether the series converges. k² + 2/ k³ - 7

After considering the given data we conclude that the distance between stone and ground level [tex]h(t) = -4.9t^2 + 900[/tex], time taken for the stone to reach the ground 18.22 seconds,the velocity with which it strikes the ground 178.76 m/s, if thrown with a down ward speed of 3m/s then the duration needed is 18.47 seconds.

A stone is dropped from the upper observation deck of a tower, 900 m above the ground. We can use the kinematic equations of motion to answer the following questions:
a) The distance of the stone above ground level at time t can be found using the equation:
[tex]h(t) = -1/2gt^2 + v_0t + h_0[/tex]
where g is the acceleration due to gravity (9.8 m/s²), v0 is the initial velocity (0 m/s), h0 is the initial height (900 m), and t is the time elapsed. Plugging in the values, we get:
[tex]h(t) = -4.9t^2 + 900[/tex]
b) To find how long it takes for the stone to reach the ground, we need to find the time when h(t) = 0:
[tex]-4.9t^2 + 900 = 0[/tex]
Solving for t, we get:
[tex]t = \sqrt(900/4.9) = 18.22 seconds[/tex]
Therefore, it takes the stone 18.22 seconds to reach the ground.
c) To find the velocity with which the stone strikes the ground, we can use the equation:
[tex]v(t) = -gt + v_0[/tex]
where v(t) is the velocity at time t. Plugging in the values, we get:
[tex]v(t) = -9.8(18.22) + 0 = -178.76 m/s[/tex]
Therefore, the stone strikes the ground with a velocity of 178.76 m/s.
d) If the stone is thrown downward with a speed of 3 m/s, we can use the same equation [tex]v(t) = -9.8(18.22) + 0 = -178.76 m/s[/tex] to find how long it takes to reach the ground. This time, [tex]v_0[/tex] is -3 m/s (since it is thrown downward) and [tex]h_0[/tex] is still 900 m. Plugging in the values, we get:
[tex]-4.9t^2 - 3t + 900 = 0[/tex]
Solving for t, we get:
t = 18.47 seconds
Therefore, it takes the stone 18.47 seconds to reach the ground when thrown downward with a speed of 3 m/s.
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The probability that the coach selects at least 3 passing plays and at least 2 running plays out of 8 plays from the playbook is approximately 0.4914 or 49.14%. This means there is a 49.14% chance of the coach choosing a combination that meets the given criteria.

To calculate the probability of the coach selecting at least 3 passing plays and at least 2 running plays out of 8 plays, we need to consider different combinations that satisfy these conditions.

1: Determine the total number of possible combinations of 8 plays from a playbook of 28 plays (16 passing plays + 12 running plays).

Total Combinations = C(28, 8) = 28! / (8! * (28-8)!) = 3,395,685

2: Calculate the number of combinations that have at least 3 passing plays and at least 2 running plays.

First, we calculate the number of combinations with exactly 3 passing plays and 2 running plays:

Number of Combinations with 3 passing and 2 running = C(16, 3) * C(12, 2) = (16! / (3! * (16-3)!) * (12! / (2! * (12-2)!) = 560 * 66 = 36,960

Next, we calculate the number of combinations with exactly 4 passing plays and 2 running plays:

Number of Combinations with 4 passing and 2 running = C(16, 4) * C(12, 2) = (16! / (4! * (16-4)!) * (12! / (2! * (12-2)!) = 1,820 * 66 = 120,120

Finally, we calculate the number of combinations with 5 passing plays and at least 2 running plays:

Number of Combinations with 5 passing and 2 or more running = C(16, 5) * (C(12, 2) + C(12, 3) + C(12, 4) + C(12, 5) + C(12, 6) + C(12, 7) + C(12, 8)) = (16! / (5! * (16-5)!) * (C(12, 2) + C(12, 3) + C(12, 4) + C(12, 5) + C(12, 6) + C(12, 7) + C(12, 8)) = 4368 * (66 + 220 + 495 + 792 + 924 + 792 + 495) = 4368 * 3786 = 16,530,048

Total Number of Combinations with at least 3 passing and 2 running plays = Number of Combinations with 3 passing and 2 running + Number of Combinations with 4 passing and 2 running + Number of Combinations with 5 passing and 2 or more running = 36,960 + 120,120 + 16,530,048 = 16,687,128

3: Calculate the probability.

Probability = (Number of Combinations with at least 3 passing and 2 running plays) / (Total Combinations) = 16,687,128 / 3,395,685 ≈ 0.4914

Therefore, the probability that the coach selects at least 3 passing plays and at least 2 running plays out of 8 plays is approximately 0.4914 or 49.14%.

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Answer:

h = 7.832

Step-by-step explanation:

This is a right angled triangle so, taking 50 as reference angle,

hypotenuse = ?

perpendicular = 6

The ratio for p and h is given by

Sin 50 = p/h

Sin 50 = 6 /h

h = 6 / Sin 50

h = 7.832

Answer:

Well you just have to follow the formula to find the distance between 2 linear plots.

d= √(x2−x1)^2+(y2−y1)^2

Step-by-step explanation:

If you follow it the answer would be

D=50

(a) Mean = (9 + 15 + 18 + 11 + 12) / 5 = 65 / 5 = 13

(b) Mean of the sample = (9 + 15) / 2 = 24 / 2 = 12

(c) The means of the samples vary, but they tend to be close to the population mean of 13.

(d) Since the problem does not provide the values for the 25 x values, we cannot construct a density histogram or determine if most of the values are near the population mean.

(a) The mean of the population is calculated by summing all the values and dividing by the total number of values:

Mean = (9 + 15 + 18 + 11 + 12) / 5 = 65 / 5 = 13

(b) To calculate the mean for a random sample of size 2, we randomly select two values from the population and calculate their mean:

Random sample: 9, 15

Mean of the sample = (9 + 15) / 2 = 24 / 2 = 12

(c) Repeatedly selecting random samples of size 2 and calculating the mean for each sample:

Here are the means for 25 random samples of size 2 (selected without replacement):

Sample 1: 9, 18 -> Mean = (9 + 18) / 2 = 27 / 2 = 13.5

Sample 2: 11, 9 -> Mean = (11 + 9) / 2 = 20 / 2 = 10

Sample 3: 15, 12 -> Mean = (15 + 12) / 2 = 27 / 2 = 13.5

...

Sample 25: 12, 15 -> Mean = (12 + 15) / 2 = 27 / 2 = 13.5

The means of the samples vary, but they tend to be close to the population mean of 13.

(d) Constructing a density histogram using the 25 x values:

Since the problem does not provide the values for the 25 x values, we cannot construct a density histogram or determine if most of the values are near the population mean.

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Answer:

(f + g)(2) = 16

(f - g)(2) = 0

(fg)(2) = 64

Step-by-step explanation:

f(2) = 8          g(2) = 8

(f + g)(2) = f(2) + g(2) = 8 + 8 = 16

(f - g)(2) = f(2) - g(2) = 8 - 8 = 0

(fg)(2) = f(2)g(2) = 8(8) = 64

Answer:

Use the degree and the leading coefficient to determine the behavior.

Falls to the left and falls to the right

Step-by-step explanation:

Given system of differential equation is(D − 1)x+ (D² + 1)y = 1 ...

(i)(D² − 1)x+ (D + 1)y = 2 ...(ii)By using systematic elimination method, we have(D²+1)(D²−1)x+(D+1)(D−1)y=D²+1×1-(D+1)×1=0Simplifying the above equation, we get(D⁴-1)x=-(D-1)y...(iii)Applying D on both sides of (iii), we get D(D⁴-1)x=-(D-1 )DyD⁵x- Dx=-(Dy-y)or D⁵x+Dy=y ... (iv)Now applying D on (i), we get(D−1)Dx+(D²+1)Dy=0or D(D²+1)y=(1-D)x ...(v)Now applying D on (ii), we get(D²−1)Dx+(D+1)Dy=0or D(D+1)x=(1+D)y ...(vi)Now, substituting the value of x and y from equations (v) and (vi) in equation (iv), we getD⁵x+(1+D)Dx=(1-D)Dy D⁵x+(1+D)Dx=-(1-D)x ...(vii) Simplifying the above equation, we getD⁶x+2D⁴x+D²x+x=0or D²(D⁴+1)x+D²x=-x ...(viii)or D²(D⁴+2)x=-xor D⁴x+2x=-xor D⁴x=-3xNow using D on both sides, we get D⁵x=-3Dxor D⁶x=-3D²x

Now, substituting the value of D²x from equation (iii) in equation (i), we get(D-1)x+(D²+1)y=1 ...(i)⇒ (D-1)x+y=1 ...(ix)Now, substituting the value of D²x from equation (iii) in equation (ii), we get(D²-1)x+(D+1)y=2 ...(ii)⇒ -(D+1)x+y=0or (D+1)x-y=0 ...(x)From equation (ix) and (x), we have2x=1or x=1/2Now, substituting the value of x in equation (ix), we have D(1/2)+y=1or y=1-1/2=1/2Thus, the solution of the given system of differential equation is(x(t), y(t))=(e^(-t/2))[(-5/3)cos((sqrt(47)/2)t)-(125/3)sin((sqrt(47)/2)t)]+(20/3)cos(3t)+(20/3)sin(3t), (1/2)

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First. Finding the x-intercepts of  

Let  be the change in water level. So to find the x-intercepts of this function we can use The Rational Zero Test that states:

To find the zeros of the polynomial:

We use the Trial-and-Error Method which states that a factor of the constant term:

can be a zero of a polynomial (the x-intercepts).

So let's use an example: Suppose you have the following polynomial:

where the constant term is . The possible zeros are the factors of this term, that is:

.

Thus:

From the foregoing, we can affirm that  are zeros of the polynomial.

Second. Construction a rough graph of  

Given that this is a polynomial, then the function is continuous. To graph it we set the roots on the coordinate system. We take the interval:

and compute  where  is a real number between -2 and -1. If , the curve start rising, if not, the curve start falling. For instance:

Therefore the curve start falling and it goes up and down until  and from this point it rises without a bound as shown in the figure below

According to the question the following using the Sampling Distribution are as follows :

1. Study Activity 1 on p.301. Then complete the following using the Sampling Distribution of a Sample Proportion web app.

i. Simulate taking a random sample of 100 voters from a large population of voters of whom 54\% voted for Brown, and record the number out of 100 that voted for Brown.

ii. Report the proportion of your sample that voted for Brown

iii. Insert below the Data Distribution generated by the web app.

i. Simulate taking a random sample of 100 voters from a large population of voters of whom 54\% voted for Brown, and record the number out of 100 that voted for Brown.

ii. Report the proportion of your sample that voted for Brown: [Insert the proportion value here]

iii. Insert below the Data Distribution generated by the web app:

[Insert the data distribution plot here]

Note: The actual values for the proportion and data distribution will vary based on the simulation results obtained from the Sampling Distribution of a Sample Proportion web app.

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Answer:

Volume = 525 in³

Step-by-step explanation:

Volume = 25 x 7 x 6 x 0.5 = 525 in³

Lagrangian interpolation is used to approximate the value of e¹.3 using three known values: eº = 1, el = 2.7183, and e² = 7.3891. We can find the minimum and maximum estimates for the error.

To approximate e¹.3 using Lagrangian interpolation, we construct a polynomial that passes through the three given points: (0, 1), (1, 2.7183), and (2, 7.3891). We can then evaluate this polynomial at x = 1.3 to estimate the value of e¹.3.

Using Lagrangian interpolation, the polynomial P(x) is given by:

P(x) = 1 * L₀(x) + 2.7183 * L₁(x) + 7.3891 * L₂(x),

where L₀(x), L₁(x), and L₂(x) are the Lagrange basis polynomials associated with the three data points.

To find the minimum and maximum estimates for the error, we need to determine the upper bound for the error term in the Lagrangian interpolation formula. The error term is given by:

E(x) = f(x) - P(x),

where f(x) is the actual function we are approximating (in this case, e^x).

To find the upper bound for the error, we can use the maximum value of the absolute value of the n+1st derivative of f(x) in the interval containing the data points.

By calculating the upper bound for the error, we can compare it to the actual error by evaluating the actual function e¹.3 and subtracting the approximation P(1.3) obtained from Lagrangian interpolation.

By analyzing the error estimates and comparing them to the actual error, we can assess the accuracy of the Lagrangian interpolation approximation in this particular case.

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By applying the initial conditions and inverse Laplace transforming, we can obtain the solution x(t) = (1 - cos(2t))u(t-5) + (3 sin(t-5))u(t-5), where u(t) is the unit step function. Therefore, the solution to the initial-value problem is x(t) = (1 - cos(2t))u(t-5) + (3 sin(t-5))u(t-5)

Taking the Laplace transform of the given differential equation x" + 4 = f(t), we obtain the algebraic equation in the Laplace domain: s^2X(s) + 4sX(s) + 4 = F(s), where X(s) is the Laplace transform of x(t) and F(s) is the Laplace transform of f(t).

Next, applying the initial conditions x(0) = 0 and x'(0) = 0, we get X(0) = 0 and sX(0) = 0. Substituting these initial conditions into the Laplace domain equation, we have s^2X(s) + 4sX(s) + 4 = F(s), with X(0) = 0 and sX(0) = 0.

Now, let's consider the Laplace transform of f(t) using the given piecewise function. For [tex]t < 5, f(t) = t^2/5, and for t > 5, f(t) = 3sin(t-5).[/tex]Taking the Laplace transform of f(t) in each interval, we have [tex]F(s) = (1/s^3) + (3/s^2) for t < 5 and F(s) = (3/s^2) * (1/(s^2+1)) for t > 5.[/tex]

Substituting these Laplace transforms into the equation[tex]s^2X(s) + 4sX(s) +[/tex]4 = F(s), we can solve for X(s). Simplifying, we obtain [tex]X(s) = (1/s^3) + (3/s^2) / (s^2 + 4s + 4) + (3/s^2) * (1/(s^2+1)).[/tex]

To find the inverse Laplace transform of X(s), we can split it into partial fractions and apply the inverse Laplace transform formula. The solution is x(t) = (1 - cos(2t))u(t-5) + (3 sin(t-5))u(t-5), where u(t) is the unit step function.

Therefore, the solution to the initial-value problem is x(t) = (1 - cos(2t))u(t-5) + (3 sin(t-5))u(t-5), where u(t) is the unit step function that ensures the piecewise function is activated at t = 5.

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The value that verifies 5p (1)-(1)-500 is -124.04

To approximate the value of the function f at x = 1 using the Taylor polynomial Pₙ = (-10)^n/ n! about x = 0, we need to find the value of P₅(1).

First, let's compute the derivatives of f(x) = e^x up to the fifth derivative:

f'(x) = e^x

f''(x) = e^x

f'''(x) = e^x

f''''(x) = e^x

f⁽⁵⁾(x) = e^x

Now, let's evaluate these derivatives at x = 0:

f(0) = e^0 = 1

f'(0) = e^0 = 1

f''(0) = e^0 = 1

f'''(0) = e^0 = 1

f''''(0) = e^0 = 1

f⁽⁵⁾(0) = e^0 = 1

Using these values, we can compute the Taylor polynomial P₅(x):

P₅(x) = f(0) + f'(0)(x - 0) + f''(0)(x - 0)²/2! + f'''(0)(x - 0)³/3! + f''''(0)(x - 0)⁴/4! + f⁽⁵⁾(0)(x - 0)⁵/5!

P₅(x) = 1 + 1x + 1x²/2! + 1x³/3! + 1x⁴/4! + 1x⁵/5!

Now, let's evaluate P₅(1):

P₅(1) = 1 + 1(1) + 1(1)²/2! + 1(1)³/3! + 1(1)⁴/4! + 1(1)⁵/5!

P₅(1) = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120

P₅(1) = 227/120

Therefore, the value that verifies 5P₅(1) - (1) - 500 is:

5P₅(1) - (1) - 500 = 5 * (227/120) - 1 - 500

= 1135/120 - 1 - 500

= 1135/120 - 120/120 - 60000/120

= (1135 - 120 - 60000)/120

= -59485/120

= -124.04

So, the value that verifies 5P₅(1) - (1) - 500 is approximately -124.04.

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Answer:

d = 20

Step-by-step explanation:

90-19 = 71

(4d-9) = 71

4d = 80

d = 20

Answer:

d = 20

Step-by-step explanation:

Complementary angles are two angles that add up to 90°.

We know that one angle is 19° and the other is (4d − 9)°. So, we can set up the equation:

19 + (4d − 9) = 90.

Solving for d, we get:

19 + (4d − 9) = 90

19 + 4d − 9 = 90

4d + 10 = 90

4d = 80

d = 20

Therefore, the value of d is 20.

Answer:

CABD

Step-by-step explanation:

Answer:

B and D

Explanation:

Answer:

x = 66

Step-by-step explanation:

m<5 + m<6 = 180

2x + 48 = 180

2x = 132

x = 66

Answer:

I feel like the answers are 10 and -2

Step-by-step explanation:

Hope this helps! Have a great day and good luck! :)

Step-by-step explanation:

[tex] \frac{12 \times 8}{2} + {( \frac{8}{2}) }^{2} \pi = \\ = 48 + 16\pi[/tex]

98.24

Answer:

17

Step-by-step explanation:

The lower quartile Q₁ is positioned at the left side of the box.

The value at the left is Q₁ is 17

Answer:

$0.24 per orange

Step-by-step explanation:

$3.84 / 16 = 0.24

The probability that at least 5 of the 8 adults say they get their financial advice from family members can be calculated using binomial probability distribution.

Formula for binomial probability distribution is:P(X=k) = nCk * pk * (1-p)n-kwhere, P(X=k) is the probability of k successes in n independent trials, p is the probability of success in one trial, q=1-p is the probability of failure in one trial, nCk is the combination of k successes in n independent trials.

In this case, the probability of success is p=0.4 as 40% of adults in the US say they get their financial advice from family members.

Therefore, the probability of failure is q=1-0.4=0.6.So, P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)P(X=k) = nCk * pk * (1-p)n-kwhere, n=8, p=0.4, q=0.6For k = 5, nCk = 8C5P(X = 5) = 8C5 * (0.4)5 * (0.6)3= 0.2787

For k = 6, nCk = 8C6P(X = 6) = 8C6 * (0.4)6 * (0.6)2= 0.1960For k = 7, nCk = 8C7P(X = 7) = 8C7 * (0.4)7 * (0.6)1= 0.0575For k = 8, nCk = 8C8P(X = 8) = 8C8 * (0.4)8 * (0.6)0= 0.0030Therefore,P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) = 0.2787 + 0.1960 + 0.0575 + 0.0030 = 0.5352

Hence, the probability that at least 5 of the 8 adults say they get their financial advice from family members is 0.5352.

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Given information that 40% of adults in the US say they get their financial advice from family members. We have to find the probability of at least 5 of the 8 say they get their financial advice from family members.

Hence, the required probability is 0.7530.

The probability of getting financial advice from family members is 40%. Let X be the number of people out of 8, who get their financial advice from family members. Here, X follows a binomial distribution with parameters n = 8 and p = 0.4. The probability of getting atleast 5 people getting their financial advice from family members is

P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)

Using binomial distribution formula, we get

P(X = x) = ${n\choose x}p^xq^{n-x}$

Where, n = 8, p = 0.4, q = 0.6

The probability of getting exactly 5 people out of 8 getting their financial advice from family members

P(X = 5) = ${8\choose 5} 0.4^5 (0.6)^{8-5}$

= 0.27869

The probability of getting exactly 6 people out of 8 getting their financial advice from family members

P(X = 6) = ${8\choose 6} 0.4^6 (0.6)^{8-6}$

= 0.29360

The probability of getting exactly 7 people out of 8 getting their financial advice from family members

P(X = 7) = ${8\choose 7} 0.4^7 (0.6)^{8-7}$

= 0.16493

The probability of getting exactly 8 people out of 8 getting their financial advice from family members

P(X = 8) = ${8\choose 8} 0.4^8 (0.6)^{8-8}$

= 0.01678

Therefore, the probability of getting at least 5 people getting their financial advice from family members is

P(X ≥ 5) = 0.27869 + 0.29360 + 0.16493 + 0.01678

= 0.7530

Hence, the required probability is 0.7530.

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Answer:

s = 3

Step-by-step explanation:

The volume formula for a cube is V = s^3, where “s” is the edge length. Since we know the volume and need to find “s,” we just do the inverse operation for an exponent, which is a radical. The cubed root of 27 is 3, so there’s your answer! Hope this is helpful & accurate. Best wishes.

To make cookies for 48 people, the recipe requires 6 cups of flour.

The oatmeal raisin cookie recipe serves 36 people and requires 2 eggs and 2 cups of flour. Since we need to make cookies for 48 people, we can calculate the amount of flour required as follows:

36 people → 2 cups of flour

1 person → (2 cups of flour) / (36 people) = (1/18) cups of flour

To make cookies for 48 people:

48 people × (1/18) cups of flour = 2.67 cups of flour

Therefore, to make cookies for 48 people, we need approximately 2.67 cups of flour.

Note: Since the result is a fraction of a cup, it is advisable to round up to the nearest whole number, so in this case, 3 cups of flour would be needed.

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Answer:

y = 15x

Step-by-step explanation:

For every flower pot purchased (y), the quantitity of the price (x) will go up by $15.

Answer:

y = 15x

Step-by-step explanation:

Answer:

V=385 cubic units

Step-by-step explanation:

The volume of a rectangular prism is given by the formula [tex]V=lwh[/tex], where l is the length, w is the width, and h is the height. Our dimensions are 5, 7, and 11. So, we have to multiply them to find the volume.

5 × 7 × 11 = 385

Thus, the volume of the rectangular prism is 385 cubic units.

The solution to the given differential equation with the initial conditions y(0) = 1 and y'(0) = 1 is:

[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]

To solve the given differential equation using Laplace transform, we will apply the Laplace transform to both sides of the equation and then solve for the transformed variable.

Let's denote the Laplace transform of y(t) as Y(s).

Taking the Laplace transform of both sides of the differential equation, we get:

[tex]s^2Y(s) + 2sY(s) + Y(s) = (s^2 + 2s + 1)/(s^2 + 4)[/tex]

Now, let's solve for Y(s):

[tex]Y(s)(s^2 + 2s + 1) = (s^2 + 2s + 1)/(s^2 + 4)\\Y(s) = (s^2 + 2s + 1)/(s^2 + 4)(s^2 + 2s + 1)[/tex]

Factoring the denominator:

[tex]Y(s) = (s^2 + 2s + 1)/((s + 1)^2(s^2 + 4))[/tex]

Now, we need to decompose the fraction into partial fractions. Let's express the numerator in terms of A, B, C, and D:

[tex]s^2 + 2s + 1 = A/(s + 1) + B/(s + 1)^2 + (Cs + D)/(s^2 + 4)[/tex]

To find the values of A, B, C, and D, we can equate the numerators:

[tex]s^2 + 2s + 1 = A(s + 1)(s^2 + 4) + B(s^2 + 4) + (Cs + D)(s + 1)^2[/tex]

Expanding and equating coefficients:

[tex]s^2 + 2s + 1 = A(s^3 + 5s^2 + 4s) + B(s^2 + 4) + (C(s^2 + 2s + 1) + D(s + 1)^2)[/tex]

Simplifying:

[tex]s^2 + 2s + 1 = (A + C)s^3 + (5A + C + D)s^2 + (4A + 2C + D)s + (4A + D)[/tex]

Equating coefficients:

A + C = 0 (coefficient of [tex]s^3[/tex])

5A + C + D = 1 (coefficient of [tex]s^2)[/tex]

4A + 2C + D = 2 (coefficient of s)

4A + D = 1 (constant term)

Solving these equations simultaneously, we find A = -1/10, B = 11/10, C = 1/10, and D = 3/10.

Now, substituting these values back into Y(s):

[tex]Y(s) = (-1/10)/(s + 1) + (11/10)/(s + 1)^2 + (1/10)(s + 3)/(s^2 + 4) + (3/10)/(s^2 + 4)[/tex]

To find y(t), we need to take the inverse Laplace transform of Y(s). Fortunately, we can use a Laplace transform table to find the inverse Laplace transform of each term.

The inverse Laplace transform of (-1/10)/(s + 1) is [tex]-e^{-t}/10.[/tex]

The inverse Laplace transform of (11/10)/(s + 1)² is (11/10)t*[tex]e^{-t}.[/tex]

The inverse Laplace transform of (1/10)(s + 3)/(s² + 4) is (1/10)cos(2t).

The inverse Laplace transform of (3/10)/(s² + 4) is (3/10)sin(2t).

Combining these results, the solution y(t) is:

[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]

Therefore, the solution to the given differential equation with the initial conditions y(0) = 1 and y'(0) = 1 is:

[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]

To learn more about Laplace transform visit:

brainly.com/question/14487937

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