Consider the system: X' X+ 13 are fundamental solutions of the corresponding homogeneous system. Find a particular solution X, = pū of the system using the method of variation of parameters.

The solution of the given differential equation by undetermined coefficients is y(x) = c1*\(e^{6x\) + c2*x*\(e^{6x\) + x + 1.

To solve the given differential equation y'' - 12y' + 36y = 36x + 7 using undetermined coefficients, follow these steps:

1. Identify the complementary function by solving the homogeneous equation y'' - 12y' + 36y = 0. The characteristic equation is \(r^2\) - 12r + 36 = 0, which factors to \((r-6)^2\) = 0. Since there is a repeated root, r = 6, the complementary function is yc(x) = c1*\(e^{6x}\) + c2*x*\(e^{6x}\).

2. Now we need to find the particular solution, yp(x). Since the nonhomogeneous term is 36x + 7, we can assume the particular solution is of the form yp(x) = Ax + B.

3. Calculate yp'(x) = A and yp''(x) = 0.

4. Substitute yp(x), yp'(x), and yp''(x) into the given differential equation: 0 - 12A + 36(Ax + B) = 36x + 7.

5. Equate the coefficients of the powers of x:
  For \(x^1\): 36A = 36, so A = 1.
  For \(x^0\): -12A + 36B = 7, so -12(1) + 36B = 7, which gives B = 1.

6. The particular solution is yp(x) = x + 1.

7. Combine the complementary function and the particular solution to get the general solution: y(x) = c1*\(e^{6x\) + c2*x*\(e^{6x}\) + x + 1.

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Answer: y = 3x+9

Step-by-step explanation:

points are -2,3 and -1,6

6-3/-1-(-2) = 3

y-int is 9 since its given

Answer:

48 degrees

Step-by-step explanation:

Answer:

Shannon run 1.5 km more on Friday than on Saturday.

Step-by-step explanation:

From the given table

i.e.

Friday run - Saturday run =  2 - 0.5

                                          = 1.5 km

Thus, Shannon run 1.5 km more on Friday than on Saturday.

Answer: x=5

Step-by-step explanation:

Answer:

Step-by-step explanation:

5x+5=x+25

Or,6x=30

Or,x=5

Answer:

It's not 2 sorry about that

Step-by-step explanation:

Answer:

32

Step-by-step explanation:

What can be represented by x,

and what divided by 8 is equal to 4 can be represented by:

x/8 = 4

×8 ×8

x = 32

The probability of finding a sample mean (Y¯) of 1.5 ounces or more when the liquor tax has no impact on consumption (µ = 0). To determine this probability, we would use the Central Limit Theorem and the z-score formula.

Since the population mean (µ) is 0, we'll need to know the population standard deviation (σ) and the sample size (n) to proceed. Without these values, it's impossible to provide an exact probability. However, I can explain the general process.

First, you would calculate the standard error (SE) using the formula SE = σ / √n. Next, you would find the z-score, which is the difference between the sample mean (Y¯) and the population mean (µ) divided by the standard error: z = (Y¯ - µ) / SE.

Once you have the z-score, you can look it up in a standard normal distribution table or use a calculator to find the probability associated with it. In this case, you're looking for the probability of finding a sample mean of 1.5 ounces or more, which corresponds to the area to the right of the z-score in the distribution.

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The slope of this situation is equal to 5.48.

In Mathematics, the slope-intercept form of the equation of a straight line is given by this mathematical expression;

y = mx + c

Where:

Based on the information, an equation that models the situation is given by this mathematical expression;

y = mx + c

y = 5.48x + 3.50

In conclusion, we can reasonably infer and logically deduce that the slope is 5.48 and the y-intercept is equal to 3.50.

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Answer:

Kate is 3 years old, Jane is 9 years old

Step-by-step explanation:

1.) First, assign variables to all of their ages. If we say Kate's age is x, Jane's age is 3 times this, which can be written as j, is 3x.

2.) Jane's age is also 2 less than twice of Kate's in 5 years. This means that her age is also 2(x+5) - 2 = j + 5. With a little simplification, you get that 2x + 8 = j + 5.

3.) Since j, Jane's age, is also 3x, we can substitute 3x in for j in the second equation. If you do this, you get 2x + 8 = 3x + 5.

4.) By moving the x onto one side and the numbers onto another, you get x = 3. X was Kate's age, meaning that Kate is 3 years old.

5.) Finally, since Jane's age is 3 times Kate's age, Jane's age is 3 * 3, which is 9. Jane is 9 years old.

Jane is 9 years old and Kate is 3 years old.To solve the problem, let's first establish variables for Jane and Kate's ages.

Let J represent Jane's age and K represent Kate's age.
According to the student question, Jane is 3 times as old as Kate, which can be represented as:
J = 3K
In 5 years, Jane's age will be 2 less than twice Kate's age, which can be represented as:
J + 5 = 2(K + 5) - 2
Now we can solve the equations step by step:

Substitute the first equation into the second equation to eliminate one of the variables:
3K + 5 = 2(K + 5) - 2
Distribute the 2 on the right side of the equation:
3K + 5 = 2K + 10 - 2
Simplify the equation by combining like terms:
3K + 5 = 2K + 8
Move the 2K term to the left side of the equation:
K = 3
now we know that Kate is currently 3 years old.
Substitute K's value back into the first equation to find Jane's age:
J = 3K
J = 3(3)
Simplify to find Jane's age:
J = 9
So, Jane is currently 9 years old.
Jane is 9 years old and Kate is 3 years old.

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Answer:

w = \(\frac{p -2l}{2}\)

Step-by-step explanation:

p = 2l + 2w  Subtract 2l from both sides of the equation

p - 2l = 2p  Divide both sides by 2

\(\frac{p -2l}{2}\) = w

ASA: ∆ABC and ∆DEF are congruent if angle B is congruent to angle E. SAS: ∆ABC and ∆DEF are congruent if angle A is congruent to angle D and AB=DE.

What is congruent ?

In geometry, congruent refers to the condition where two or more figures have the same size and shape. Congruent figures have the same dimensions, angles, and proportions. Two figures are congruent if they can be superimposed exactly on each.

1) If, C.) Angle B is congruent to angle E then ∆ABC and ∆DEF are congruent by the ASA criterion.

2) If, D.) Angle A is congruent to angle D then ∆ABC and ∆DEF are congruent by the SAS criterion.

3) ∆ABC and ∆DEF are congruent if B.) AB=DE

ASA criterion: If in two triangles, two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

ASA: ∆ABC and ∆DEF are congruent if angle B is congruent to angle E. SAS: ∆ABC and ∆DEF are congruent if angle A is congruent to angle D and AB=DE.

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The probability of selecting a bird would be = 2/15

Probability is defined as the ability of an outcome to occur by chance or not.

The formula that can be used to calculate the outcome of an event= P(E) = number of favorable outcomes/Total number of outcomes

Where, P(E) is the probability of any event.

Let E be an event of selecting an insect.

According to the given question

We have

Total number of insects = 10

Total number of trees = 8

Total number of flowers = 8

Total number of birds = 4

Now, the total number of outcomes = 10 + 8 + 8 + 4 = 30

Total number of favorable outcomes(birds) = 4

Therefore,

The probability of selecting a bird = 4/30 = 2/15

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Complete question:

A stack of 30 science flashcards includes a review card for each of the

following: 10 insects, 8 trees, 8 flowers, and 4 birds.

What is the probability of randomly selecting a bird.

Minimization of  f(x) = |x+3| + x^3 at the endpoints (-2 and 6)  the minimum value of the function is approximately 3.84, which occurs at x= \sqrt{1/3}

within the given interval.

To minimize the function subject to the constraint f(x) = |x+3| + x^3 that x lies in the interval [-2, 6], we need to find the value of x that minimizes f(x) within that interval.

First, let's analyze the function f(x). The absolute value term |x+3| can be rewritten as:

|x+3| =

x+3 if x+3 >= 0

-(x+3) if x+3 < 0

Since the interval [-2, 6] includes both positive and negative values of x+3, we need to consider both cases.

Case 1: x+3 >= 0

In this case, f(x) = (x+3) + x^3 = 2x + x^3 + 3

Case 2: x+3 < 0

In this case, f(x) = -(x+3) + x^3 = -2x + x^3 - 3

Now, we can find the minimum of f(x) within the given interval by evaluating the function at the endpoints (-2 and 6) and at any critical points within the interval.

Calculating the values of f(x) at x = -2, 6, and the critical points, we can determine the minimum value of f(x) and the corresponding value of x.

Since the equation involves both absolute value and a cubic term, it is not possible to find a closed-form solution or an exact minimum value without numerical methods or approximation techniques.

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Answer:

(6, 12)

Step-by-step explanation:

{ 3x - y = 6

{ x - 2y = -18

by using elimination:

6x -2y=12

x - 2y= -18

________-

5x = 30

x = 6

y = 18-6 = 12

the solutions = {(6, 12)}

Jose has 3/5 lb

Maria = 3/5 - 1/2 = 1/10 lb

Atul has 2/3 lb

Pounds of candy Autul have than Maria:-

2/3 —3/5 = 4/15

Answer:- 4/15

Step-by-step explanation:

103° + 37° + < C = 180° { being sum of angles of triangle }

< C = 180° - 140°

< C = 40°

Hope it will help :)

Answer:

43 ; 0.88493

Step-by-step explanation:

Using the Zscore formula :

Zscore = (x - m) / s

m = mean ; s = standard deviation

Profit = $1.5 - $1.2 = $0.3

Loss = $1.2 - $1 = 0.2

Cummlative probability :

Profit / (profit + loss)

0.3 / (0.3 + 0.2) = 0.3 / 0.5 = 0.6

To obtain the x, at Z at 0.6 = 0.26

m = 40 ; s= 10

Hence,

0.26 = (x - 40) / 10

0.26 * 10 = x - 40

2.6 = x - 40

2.6 + 40 = x

x = 42.6 `; 43 approximately

Probability of meeting day's demand at Wrigley

x = 52

P(x < 52) :

Zscore = (52 - 40) / 10

P(Z < 1.2)

Z = 0.88493

The value of {x} is equivalent to {x} = 4β - (y + z).

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.

Given is that triangle and the square have equal perimeters.

Assume the lengths of the triangle are {x}, {y} and {z} respectively.

Let the side length of a square be β. We can write -

P{T} = P{S}

(x + y + z) = 4β

{x} = 4β - (y + z)

Therefore, the value of {x} is equivalent to {x} = 4β - (y + z).

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The z-score for X = 44 is approximately -1.43.

To calculate the z-score for X = 44, we need to first calculate the mean and standard deviation of the population:

Mean (μ) = (149 + 187 + 110 + 108 + 108 + 143 + 9 + 159 + 187) / 9 = 125.89

Standard deviation (σ) = sqrt([Σ(xi - μ)^2] / N) = 57.23

where:

Σ is the sum over all values

xi is the i-th value in the population

N is the total number of values in the population

Now we can calculate the z-score using the formula:

z = (X - μ) / σ

Substituting the given values, we get:

z = (44 - 125.89) / 57.23 ≈ -1.43 (rounded to 2 decimal places)

Therefore, the z-score for X = 44 is approximately -1.43.

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Answer:

I'm pretty sure it's C I may be wrong though

Step-by-step explanation:

The general solution to the differential equation dy/dx = 1/3(sin x − xy^2), y(0)=5 is: y = ±√[(sin x - e^(x/2)/25)/x], if sin x - xy^2 > 0 and y(0) = 5

To solve this differential equation, we can use separation of variables.

First, we can rearrange the equation to get dy/dx on one side and the rest on the other side:

dy/dx = 1/3(sin x − xy^2)

dy/(sin x - xy^2) = dx/3

Now we can integrate both sides:

∫dy/(sin x - xy^2) = ∫dx/3

To integrate the left side, we can use substitution. Let u = xy^2, then du/dx = y^2 + 2xy(dy/dx). Substituting these expressions into the left side gives:

∫dy/(sin x - xy^2) = ∫du/(sin x - u)

= -1/2∫d(cos x - u/sin x)

= -1/2 ln|sin x - xy^2| + C1

For the right side, we simply integrate with respect to x:

∫dx/3 = x/3 + C2

Putting these together, we get:

-1/2 ln|sin x - xy^2| = x/3 + C

To solve for y, we can exponentiate both sides:

|sin x - xy^2|^-1/2 = e^(2C/3 - x/3)

|sin x - xy^2| = 1/e^(2C/3 - x/3)

Since the absolute value of sin x - xy^2 can be either positive or negative, we need to consider both cases.

Case 1: sin x - xy^2 > 0

In this case, we have:

sin x - xy^2 = 1/e^(2C/3 - x/3)

Solving for y, we get:

y = ±√[(sin x - 1/e^(2C/3 - x/3))/x]

Note that the initial condition y(0) = 5 only applies to the positive square root. We can use this condition to solve for C:

y(0) = √(sin 0 - 1/e^(2C/3)) = √(0 - 1/e^(2C/3)) = 5

Squaring both sides and solving for C, we get:

C = 3/2 ln(1/25)

Putting this value of C back into the expression for y, we get:

y = √[(sin x - e^(x/2)/25)/x]

Case 2: sin x - xy^2 < 0

In this case, we have:

- sin x + xy^2 = 1/e^(2C/3 - x/3)

Solving for y, we get:

y = ±√[(e^(2C/3 - x/3) - sin x)/x]

Again, using the initial condition y(0) = 5 and solving for C, we get:

C = 3/2 ln(1/25) + 2/3 ln(5)

Putting this value of C back into the expression for y, we get:

y = -√[(e^(2/3 ln 5 - x/3) - sin x)/x]

So the general solution to the differential equation dy/dx = 1/3(sin x − xy^2), y(0)=5 is:

y = ±√[(sin x - e^(x/2)/25)/x], if sin x - xy^2 > 0 and y(0) = 5

y = -√[(e^(2/3 ln 5 - x/3) - sin x)/x], if sin x - xy^2 < 0 and y(0) = 5

Note that there is no solution for y when sin x - xy^2 = 0.

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The linear function is given by y = (1/9)x + 7, where x is the number of minutes passed, and y is the amount of water in the bucket in inches. The linear function rule to model how many inches of water is in the bucket after any number of minutes can be determined by using the slope-intercept form of the equation of a line.

which is given by

y = mx + b, where m is the slope and b is the y-intercept. In this case, we know that water begins to drip out of a pipe into an empty bucket after 63, and there are 7 inches of water in the bucket. So, the y-intercept of the line is 7, which is the value of y when x = 0. To find the slope, we need to use the rate at which the water is dripping from the pipe. We can express this rate as the ratio of the change in y (the amount of water in the bucket) to the change in x (the number of minutes passed).

Water is dripping from a pipe into an empty bucket, and we want to determine how many inches of water are in the bucket after any number of minutes. We can model the relationship between the amount of water in the bucket and the number of minutes that have passed using a linear function. To do this, we need to find the slope and y-intercept of the line that represents this relationship.

We know that after 63 minutes, there are 7 inches of water in the bucket. This means that the y-intercept of the line is 7. To find the slope of the line, we need to use the rate at which water is dripping from the pipe. We can express this rate as the ratio of the change in y (the amount of water in the bucket) to the change in x (the number of minutes placed).

Since water is dripping out of the pipe at a constant rate, we can assume this ratio is constant. We can use the information that the bucket has 7 inches of water after 63 minutes to find this ratio. If we let y1 be the amount of water in the bucket after 63 minutes (which is 7 inches) and x1 be 63, then the ratio is given by:

= (change in y) / (change in x)

= (y1 - y0) / (x1 - x0)

= (7 - 0) / (63 - 0)

= 1/9

So, the slope of the line is 1/9. Therefore, the linear function rule to model how many inches of water is in the bucket after any number of minutes is:

y = (1/9)x + 7.

We can use a linear function to model the relationship between the amount of water in the bucket and the number of minutes that have passed. The function is given by:

y = (1/9)x + 7, where x is the number of minutes passed, and y is the amount of water in the bucket in inches. The y-intercept of the line is 7, which means that when x = 0 (i.e., at the beginning), there are 7 inches of water in the bucket. The slope of the line is 1/9, which means that for every 9 minutes that pass, 1 inch of water is added to the bucket.

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The lines that represent the approximate directrices of the ellipse are x = -6.6 and x = 10.6.

The lines that represent the approximate directrices of the ellipse are x = -6.6 and x = 10.6.

Given an ellipse with center (0,0) that has the equation

\($\frac{x^2}{225}+\frac{y^2}{400}=1$\),

find the directrices.

Solution: The standard equation of an ellipse with center (0,0) is

\($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$\)

Where 'a' is the semi-major axis and 'b' is the semi-minor axis. Comparing this equation with

\($\frac{x^2}{225}+\frac{y^2}{400}=1$\)

gives us: a=15 and b=20.

The distance between the center and each focus is given by the relation:

\($c=\sqrt{a^2-b^2}$\)

Where 'c' is the distance between the center and each focus.

Substituting the values of 'a' and 'b' gives:

\($c=\sqrt{15^2-20^2}$ = $\sqrt{-175}$ = $i\sqrt{175}$\)

The directrices are on the major axis. The distance between the center and each directrix is

\($d=\frac{a^2}{c}$\).

Substituting the value of 'a' and 'c' gives:

\(d=\frac{15^2}{i\sqrt{175}}$ $=$ $\frac{225}{i\sqrt{175}}$\)

\($= \frac{15\sqrt{7}}{7}i$\)

Therefore, the equations of the directrices are \($x=-\frac{15\sqrt{7}}{7}$\) and \($x=\frac{15\sqrt{7}}{7}$\)

Round to the nearest tenth, the answer is -6.6 and 10.6 respectively. Thus, the lines that represent the approximate directrices of the ellipse are x = -6.6 and x = 10.6.

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If the cost of one paperback book is less than $1, he would receive even more change.

To solve this problem, we need to know the cost of one paperback book. Unfortunately, that information is missing from the question. Without it, we cannot determine the total cost of the 5 paperback books and therefore cannot calculate Prakash's change.

However, we can make an educated guess based on the fact that the question mentions both paperback and hardcover books. Typically, hardcover books are more expensive than paperbacks, so we can assume that the cost of one paperback book is less than the cost of one hardcover book.

Assuming that each hardcover book costs $10, the total cost of 3 hardcover books would be $30. If Prakash gives the clerk $20 bills, he would pay $60 in total. If we subtract the cost of the 3 hardcover books ($30) from the total amount paid ($60), we are left with $30.

This is the maximum amount of change Prakash could receive if each paperback book costs $1.

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Answer:

(2x - 1)(2x + 1)

Step-by-step explanation:

Difference of Squares: (ax + b)(ax - b) = (ax² - b²)

Answer:

4×^2-1=(2× +1)(2× -1)

Step-by-step explanation:

using a^2 - b^2 =(a+b)(a-b)

Answer:

y=2x+5

Step-by-step explanation:

Let's solve for y.

−2x+y=5

Step 1: Add 2x to both sides.

−2x+y+2x=5+2x

y=2x+5

Answer:

y = 2x + 5

Step-by-step explanation:

-2x + y = 5 or y -2x = 5

We want what y equals so we need to isloate it. To do this, we need to add2x to both sides. This will cancel the 2x on the left side and now we have y alone. Remember, you have to do it to both sides. Therefore:

y = 2x + 5

Note: This equation is in slope-intercept form, y= mx + b

If this was the equation of a line:

  m (slope) = 2

  b (y-intercept) = 5

Hope this helps!

Answer:

56

20% of 70 is 14

so 14 kids canceled

so that leaves 56

Answer:

We can use the properties of logarithms to simplify this expression:

log2(9x^10/y²) = log2(9) + log2(x^10) - log2(y²)

Now we can apply the power rule of logarithms to the second term:

log2(x^10) = 10 log2(x)

Substituting back into the original expression:

log2(9x^10/y²) = log2(9) + 10 log2(x) - log2(y²)

This is the simplified form of the expression.