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The general solution for the differential equation \(x^3y''-3x^2y''+6xy'-6y=x^4 ln(x)\) is \(y=C_1x^2+C_2x+\frac{C_3}{x}\), where C₁, C₂, and C₃ are arbitrary constants.

The differential equation is Euler-Cauchy type, so we can try to solve it using the method of separation of variables. We can write the equation as:

\(y''' - \frac{3}{x} y'' + \frac{6}{x} y' - \frac{6}{x^2} y = x^2 \ln(x)\)

If we let z=y′, then we can rewrite the equation as:

\(z' - \frac{3}{x} z + \frac{6}{x} y = x^2 \ln(x)\)

Now we can separate the variables:

\(\frac{dz}{x^2} - \frac{3}{x} z = x^2 \ln(x)\)

We can integrate both sides of the equation:

\(\int \frac{dz}{x^2} - \int \frac{3}{x} z = \int x^2 \ln(x) dx\)

We can use the substitution u=x² and du=2xdx to evaluate the integral on the right-hand side:

\(\left[ -\frac{z}{x} - \frac{3}{2} z^2 \right] = \frac{2}{3} x^3 \ln(x) + \frac{C}{2}\)

Solving for z, we get: \(z = \frac{2}{3} x^3 \ln(x) + \frac{C}{2} x\)

We can then substitute back to get y′: \(y' = \frac{2}{3} x^3 \ln(x) + \frac{C}{2} x\)

Integrating both sides of the equation, we get y: \(y = \frac{2}{9} x^4 \ln(x) + \frac{C}{2} x^2 + C_1\)

where C₁ is an arbitrary constant.

Therefore, the general solution for the differential equation is y= C₁x²+C₂x+C₃/x, where C₁, C₂, and C₃ are arbitrary constants.

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Answer:

(5, 27)

Step-by-step explanation:

-3(x^2 - 10x + 16)  [Expand the equation using the FOIL method]

-3(x^2 - 10x) - 48  [Move the 16 outside of the parenthesis (16 * - 3 = -48]

-3(x^2 - 10x + 25) +75 - 48  [Complete the square]

-3(x-5)^2 + 27  [Simplify. The equation is now in vertex form.]

Answer:

4. rise

match the following definition with the correct term:

the vertical distance between two points on a line

1. run

2. rate of run

3. slope

4. rise

Answer:

Each term in pattern B is 9 less than the corresponding term in pattern A.

Answer:

The volume of a cube is equivalent to the length of its edge cubed, thus the length of the edge of the box is 3√1331 = 11 inches.

Step-by-step explanation:

sorry if its wrong :) hopes it helps

Answer:

The answer is

Step-by-step explanation:

Since the figure is a cube all it's sides are equal

Volume of a cube = l³

where

l is the length of one side

From the question

volume = 1331 in³

The length of one side is

\(1331 = {l}^{3} \\ \sqrt[3]{ {l}^{3} } = \sqrt[3]{1331} \\ \)

We have the final answer as

Hope this helps you

The volume of the solid obtained by rotating the region bounded by the curves y = 1/x5, y = 0, x = 1, and x = 8, about y = -2 is 4214.07 cubic units

To find the volume of the solid obtained by rotating the region bounded by the curves y = 1/x5, y = 0, x = 1, and x = 8, about y = -2, we need to use the method of cylindrical shells.

We can integrate the volume of the shells from x = 1 to x = 8. Since the axis of rotation is located at y = -2, we need to add 2 to the formula for the radius, which is x.

To integrate the volume of the shells, we can use the formula:

V = ∫(2πrh)dx

where r = x + 2h = 1/x5

Since the limits of integration are from x = 1 to x = 8, we have:

V = ∫1/x5(2π(x + 2))(dx)

from 1 to 8 = ∫16(2π(x + 2))(dx)

from 1 to 8= 32π∫(x + 2)(dx)

from 1 to 8= 32π[(1/2)(x² + 4x)]

from 1 to 8= 32π[(1/2)(8² + 4(8)) - (1/2)(1² + 4(1))]= 32π[128 + 12 - 2 - 4]= 32π[134]= 4214.07

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Answer:

The answer is the option A

The line is horizontal. The slope is

Step-by-step explanation:

we know that

The slope of the line parallel to the x-axis is equal to zero and is called a horizontal line

The slope of the line parallel to the y-axis is undefined and is called a vertical line

In this problem we have

This is a line parallel to the x-axis

therefore

Is a horizontal line with slope equal to zero

Answer:

219

Step-by-step explanation:

adulta 219×17=3723

5563-323=1840

1840÷10=184 kids

219+184=403

To find the series solution for the differential equation y'' + 2xy' + 2y = 0, we can assume a power series solution of the form:

Now, substitute y(x), y'(x), and y''(x) into the differential equation:

∑(n=0 to ∞) aₙn(n-1) xⁿ⁻² + 2x ∑(n=0 to ∞) aₙn xⁿ⁻¹ + 2 ∑(n=0 to ∞) aₙxⁿ = 0

We can simplify this equation by combining the terms with the same powers of x. Let's manipulate the equation step by step:

We can combine the three summations into a single summation:

∑(n=0 to ∞) (aₙ₊₂(n+1)n + 2aₙ₊₁ + 2aₙ) xⁿ = 0

Since this equation holds for all values of x, the coefficients of the terms must be zero. Therefore, we have:

This is the recurrence relation that determines the coefficients of the power series solution To find the series solution, we can start with initial conditions. Let's assume that y(0) = y₀ and y'(0) = y'₀. This gives us the following initial terms:

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Answer:b

Step-by-step explanation:

Answer:

There is 2 blue beads on Nicole's necklace.

Step-by-step explanation:

In order to find out how much blue beads Nicole has on his necklace we just have to multiply the total among of the bead on the necklace by the 10%.

20 x 10% = \((20) (\frac{1}{10} )\) = \(2\)

7/10= 28/40

5/8=25/40

0.8=4/5=32/40

32>28>25

So 0.8>7/10>5/8

The trigonometric identity that is true is:

sin G = cos I. Option D

First, we need to know that;

One acute angle in a right triangle has a sine value equal to the cosine value of the other acute angle.

From the information given, we have;

In a right scalene triangle GHI, where both angle G and angle I are acute, the value of sin G is equivalent to cos I.

Several alternatives, including sin G = sin I, cos G = cos I, tan G = tan I, sin G = tan I, sin G being the inverse of cos I, and cos G being the inverse of cos I, are invalid for a right scalene triangle with a known right angle.

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Home prices are probably skewed to the right and not symmetric. So the correct option would be A.

This makes the median a better representation of the center than the mean which would be influenced by the extremely high-priced homes.

Reporting the mean would give the impression that the​ "typical" home price is higher than it is.

they substitute the median for the mean. The median is unaffected by severe outliers or asymmetric score distributions because it only takes one or two values. In contrast, skewed distributions might have different mean and mode values.

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The distance from the center of dilation, Q, to the image of vertex A will be 4 units.

According to the given rule of dilation, D subscript Q, two-thirds, the triangle ABC will be dilated with a scale factor of two-thirds centered at point Q.

Since point Q is the center of dilation and the distance from triangle ABC to point Q is 6 units, the image of vertex A will be 2/3 times the distance from A to Q. Therefore, the distance from A' (image of A) to Q will be (2/3) x 6 = 4 units.

By applying the scale factor to the distances, we can determine that the length of A'B' is (2/3) x  3 = 2 units, the length of B'C' is (2/3) x 7 = 14/3 units, and the length of A'C' is (2/3) x 8 = 16/3 units.

Thus, the distance from the center of dilation, Q, to the image of vertex A is 4 units.

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A basis for the kernel of T is [2t, -t/2, t], where t is a parameter.

Two vectors x and y in R³ that do not belong to the kernel of T are [1, 0, 0] and [0, 1, 0].

A basis for the kernel of T is [2t, -t/2, t], where t is a parameter.

Two vectors x and y in R³ that do not belong to the kernel of T are [1, 0, 0] and [0, 1, 0].

We have,

To find a basis for the kernel of T, we need to solve the equation T(x) = 0, where x = [x₁, x₂, x₃] is a vector in R³.

From the given transformation T, we have:

T(x) = [2x₁ - (x₁ + 2x₂ + x₃), x₁ + 3x₂ + 2x₃ - (2x₁ + 5x₂ + 3x₃), 2x₁ + 5x₂ + 3x₃ - (2x₁ + 5x₂ + 3x₃)]

Simplifying further, we get:

T(x) = [x₁ - 2x₂ - x₃, -x₁ - 2x₂ - x₃, 0]

To find the kernel, we need to solve the system of equations:

x₁ - 2x₂ - x₃ = 0

-x₁ - 2x₂ - x₃ = 0

0 = 0

We can rewrite the system in augmented matrix form:

[1 -2 -1 | 0]

[-1 -2 -1 | 0]

[0 0 0 | 0]

Row reducing the augmented matrix, we get:

[1 -2 -1 | 0]

[0 -4 -2 | 0]

[0 0 0 | 0]

Simplifying further, we have:

[1 -2 -1 | 0]

[0 1/2 1/4 | 0]

[0 0 0 | 0]

From the row-reduced echelon form, we can see that the variables x₁ and x₂ are leading variables, while x₃ is a free variable.

Let x₃ = t (a parameter).

Then, we can express x₁ and x₂ in terms of x₃:

x₁ = 2t

x₂ = -t/2

Therefore, the kernel of T can be represented by the vectors [2t, -t/2, t], where t is a parameter.

Now,

To find x ‡ y in R³, we need to find two linearly independent vectors x and y that do not belong to the kernel of T.

Choosing x = [1, 0, 0] and y = [0, 1, 0], we can see that neither x nor y satisfies T(x) = 0 or T(y) = 0.

Therefore, x and y do not belong to the kernel of T.

Thus,

A basis for the kernel of T is [2t, -t/2, t], where t is a parameter.

Two vectors x and y in R³ that do not belong to the kernel of T are [1, 0, 0] and [0, 1, 0].

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The measure of angle ∠3 and angle ∠6, which are alternate interior angles formed between the parallel lines m and n are;

m∠3 = 60°, m∠6 = 60°

Alternate interior angles are angles formed in the inside part of two lines that have a common transversal, and on opposite sides of the transversal.

The information are;

Line m is parallel to line n (m ║ n)

m∠3 = 7·x - 10

m∠6 = 5·x + 10

From the possible diagram in a similar question, we get;

∠3 and ∠6 are alternate interior angles formed between parallel lines therefore;

∠3 ≅ ∠6

m∠3 = m∠6 (definition of congruent angles)

7·x - 10 = 5·x + 10 (substitution property of equality)

7·x - 5·x = 10 + 10 = 20

2·x = 20

x = 20 ÷ 2 = 10

x = 10

m∠3 = 7 × 10 - 10 = 60

m∠3 = 60°

m∠3 = m∠6

Therefore;

m∠6 = 60°

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Answer:

enter the step by step answer u did and then add the number the match and enter them in the box and u shall be done

Step-by-step explanation:

The joint pmf of X and Y is:

Px,y(0,1) = 1/4

Px,y(1,0) = 1/4

Px,y(1,1) = 1/4

Px,y(2,0) = 1/4

To find the joint probability mass function (pmf) of X and Y, we need to consider all possible outcomes of the two independent flips of a fair coin.

There are four possible outcomes:

H, H (heads on the first flip and heads on the second flip)

H, T (heads on the first flip and tails on the second flip)

T, H (tails on the first flip and heads on the second flip)

T, T (tails on the first flip and tails on the second flip)

Let's calculate the probability of each outcome first:

P(H, H) = 1/4

P(H, T) = 1/4

P(T, H) = 1/4

P(T, T) = 1/4

Now we define X as the total number of tails and Y as the number of heads on the last flip. We can calculate the values of X and Y for each outcome:

X = 0 (no tails), Y = 1 (one head on the last flip)

X = 1 (one tail), Y = 0 (no heads on the last flip)

X = 1 (one tail), Y = 1 (one head on the last flip)

X = 2 (two tails), Y = 0 (no heads on the last flip)

We can now calculate the probability of each combination of X and Y:

P(X=0, Y=1) = P(H, H) = 1/4

P(X=1, Y=0) = P(H, T) = 1/4

P(X=1, Y=1) = P(T, H) = 1/4

P(X=2, Y=0) = P(T, T) = 1/4

since the coin is fair, the probability of getting a head or a tail on each flip is 1/2.

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Let's consider all the possible outcomes of two independent flips .

The possible outcomes for X and Y are as follows:

If both flips are tails (outcome T,T), then X = 2 and Y = 0.

If the first flip is tails and the second flip is heads (outcome T,H), then X = 1 and Y = 1.

If the first flip is heads and the second flip is tails (outcome H,T), then X = 1 and Y = 0.

If both flips are heads (outcome H,H), then X = 0 and Y = 1.

For each outcome, we can calculate the joint probability as the product of the individual probabilities of each flip. For example, for the outcome T,H, the probability is P(T,H) = P(T) * P(H) = 1/4 * 1/2 = 1/8.

Using this approach, we can calculate the joint PMF for each possible value of X and Y as follows:

P(X=2, Y=0) = P(T,T) = 1/4

P(X=1, Y=1) = P(T,H) = 1/8

P(X=1, Y=0) = P(H,T) = 1/8

P(X=0, Y=1) = P(H,H) = 1/4

Therefore, the joint PMF of X and Y is given by:

Y=0 Y=1

X=0 0 1/4

X=1 1/8 0

X=2 1/4 0

This table shows the probability of each possible pair of values for X and Y. For example, P(X=1, Y=0) = 1/8, indicating that there is a 1/8 probability of getting one tail and then a head on the second flip.

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Answer:

Function 2

Step-by-step explanation:

✔️Slope of function 1:

Slope (m) = ∆y/∆x

Using two points on the line, (0, 4) and (1, 9),

Slope (m) = (9 - 4) / (1 - 0)

Slope (m) = 5/1

Slope (m) = 5

✔️Slope of function 2:

Slope (m) = ∆y/∆x

Using the coordinates of any two given pairs form the table, (0, 1) and (1, -3):

Slope (m) = (-3 - 1) / (1 - 0)

Slope (m) = -4/1

Slope (m) = -4

✔️Slope of Function 3:

The function is given in the slope-intercept form, y = mx + b

m = slope

The function y = 2x - 5, therefore has a slope (m) of 2

Slope = 2

✔️Slope of Function 4:

Slope (m) = -1

✅Thus, the functions that has slopes less than -2 are:

Function 2: slope = -4

Function 1, 3, and 4 all have slopes that are greater than -2

Answer:

It would be at the beginning of the trials

Step-by-step explanation:

The time required for the ball to reach the ground is approximately 0.56 seconds, and the horizontal displacement covered by the ball is approximately 1.34 meters.

To find the time required for the ball to reach the ground, we can use the formula for vertical motion. Since the ball is thrown horizontally, the initial vertical velocity is 0 m/s. The height of the table is 1.6 m, so we can use the formula h = (1/2)gt^2, where h is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time. Plugging in the values, we get 1.6 = (1/2)(9.8)t^2. Solving for t, we find t ≈ 0.56 seconds.

To find the horizontal displacement covered by the ball, we can use the formula d = vt, where d is the displacement, v is the initial horizontal velocity, and t is the time. Plugging in the values, we get d = 2.4 m/s * 0.56 s ≈ 1.34 meters.

Therefore, the time required for the ball to reach the ground is approximately 0.56 seconds and the horizontal displacement covered by the ball is approximately 1.34 meters.

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The tendency to perceive meaningful patterns in random sequences of outcomes often leads us to underestimate the extent to which outcomes result from CHANCE .

The word "chance" describes unpredictability or the unexpected in relation to things like events that happen without a clear reason why and without human intention.

The conclusion is that chance is the tendency of people to recognize different kinds of significant patterns in a random order or sequence in addition to evaluating any kind of outcome. Because chance can also result in an underestimating of a system's conclusion or result, it is crucial to consider it when conducting an investigation.

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The solution of the differential equation dy/dx = x/y with the given initial condition y(0) = -2 is found to be y = √(x²+4).

The differential equation that we have to solve is given to be, dy/dx = x/y.

Now, the initial condition is y(0) = -2, it means on putting x = 0 in the final answer we get y = -2. So,

dy/dx = x/y

ydy = xdx

Now, integrating both sides of the equation, ydy = xdx,

∫ydy = ∫xdx

y²/2 = x²/2 + C

y = √(x²+2C)

C is the constant of integration, Putting x = 0,

-2 = √2C

Squaring both sides,

4 = 2C

C = 2, now putting the value of C in the solution,

y = √(x²+2(2))

So, the final solution of the differential equation is y = √(x²+4).

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Complete question - Find the solution of the differential equation that satisfies the given initial condition.

dy/dx = x/y, y(0) = -2.

Answer:

What is statement and reason in geometry?

The order of the statements in the proof is not always fixed, but make sure the order makes logical sense. Reasons will be definitions, postulates, properties and previously proven theorems. “Given” is only used as a reason if the information in the statement column was told in the problem.

Step-by-step explanation:

The quarterly dividend fοr οne share οf Bοοtle is $12.52.

A divisiοn is οne οf the fundamental mathematical οperatiοns that divides a larger number intο smaller grοups with the same number οf cοmpοnents. Hοw many tοtal grοups will be established, fοr instance, if 20 students need tο be separated intο grοups οf five fοr a spοrting event?

The divisiοn οperatiοn makes it simple tο tackle such issues. Divide 20 by 5 in this case. 20 x 5 = 4 will be the οutcοme. There will therefοre be 4 grοups with 5 students each. By multiplying 4 by 5 and receiving the result 20, yοu may cοnfirm this value.

Tο find the quarterly dividend fοr οne share οf Bοοtle, we can divide the tοtal quarterly dividend payment by the number οf shares Leslie οwns:

Quarterly dividend per share = Tοtal quarterly dividend payment / Number οf shares

Quarterly dividend per share = $1,490 / 119

Quarterly dividend per share = $12.52 (rοunded tο twο decimal places)

Therefore, the quarterly dividend for one share of Bootle is $12.52.

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Answer:

the answer is A {126,168,189}

Answer: 20%

Step-by-step explanation:

Human capital refers to the expertise that people gain as a result of working. This includes skills they gain from training and actually working. For instance, the skills a Brainly website designer gets from working on Brainly.

Research suggests that this human capital has led to about a 20% growth in U.S. productivity from the year 1950 to the year 2007 which means that it is a very significant factor in the growth of the country.