Solve for the variable
Round to 3 decimal places
12
70°
у

The scalars a = 2.5 and b = 1.5 where satisfy u = a v + b w. 4v - 1w 0.40 V + 0.67w 4v + 1w 2.5 v + 1.5w.

We need to discover scalars a and b such that u = a v + b w.

We are able to set up a framework of conditions utilizing the components of the vectors:

a + b = 4 (from the i-component)

a + b = 1 (from the j-component)

solving this framework of conditions, we get:

a = 2.5

b = 1.5

Subsequently, we have:

u = 2.5v + 1.5w

Substituting the given values for v and w, we get:

u = 2.5(i + j) + 1.5(i - j)

= (2.5 + 1.5)i + (2.5 - 1.5)j

= 4i + j

So the values we found for a and b fulfill the equation u = a v + b w, and we will check that the coming about vector matches the given esteem of u.

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In correlational designs for exploring and quantifying relationships between continuous variables in a single population.

The most commonly used parametric test in a correlational design is the Pearson correlation coefficient, also known as Pearson's r or simply r. It is used to measure the strength and direction of a linear relationship between two continuous variables.

The Pearson correlation coefficient, r, ranges from -1 to 1. A value of 1 indicates a perfect positive linear relationship, a value of -1 indicates a perfect negative linear relationship, and a value of 0 indicates no linear relationship between the variables.

To use Pearson's r, the data must meet certain assumptions, including that the variables are normally distributed, there is a linear relationship between the variables, and there are no outliers or influential data points.

Once the data meets the assumptions, the Pearson correlation coefficient can be calculated using a statistical software or by hand. The resulting r value can then be interpreted and used to make conclusions about the relationship between the variables.

Overall, the Pearson correlation coefficient is a useful and commonly used tool in correlational designs for exploring and quantifying relationships between continuous variables in a single population.

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f(x) = Σ [(-1)^n * 2^(2n) * (x-4)^(2n)]/(2n)! for n=0,1,2,...

To find the Taylor series for f centered at x=4, we will use the given information about the function's derivatives at that point.

For f(2n)(4), we have:
f(2n)(4) = (-1)^n * 2^(2n)

For f(2n+1)(4), we have:
f(2n+1)(4) = 0 for all n

The Taylor series for a function f centered at x=c is given by:
f(x) = Σ [f^(n)(c) * (x-c)^n]/n! for n=0,1,2,...

In our case, c=4. Since all odd derivatives are 0, the series will only have even terms. So the Taylor series for f centered at x=4 will be:

f(x) = Σ [f(2n)(4) * (x-4)^(2n)]/(2n)! for n=0,1,2,...

Substituting the expression for f(2n)(4):

f(x) = Σ [(-1)^n * 2^(2n) * (x-4)^(2n)]/(2n)! for n=0,1,2,...

This is the Taylor series representation for f centered at x=4.

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Unit Rate: a square yard for every 100 grass seeds

A rate is a ratio used to compare two different types of quantities with different units. The unit rate, on the other hand, shows how many units of one item equate to a single unit of another quantity. When the denominator in rate is one, we call it unit rate.

Rate:

8 seeds per square foot

6 plants in 1 square yard

25 trees per 25 square yards

2 plants in a square foot

4 acres for 800 plants

Unit Rate:

a square yard for every 100 grass seeds

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An observation with an unusually large (in absolute value) positive or negative residual is classified as an outlier.

An observation with a residual refers to the difference between the observed value and the predicted value in a statistical model. Residuals are used to assess the accuracy of a model's predictions. When a residual has an unusually large value, either positive or negative, it is considered as an outlier.

An outlier is an observation that deviates significantly from the majority of the data points in a dataset. Outliers can have a significant impact on the overall results of statistical analyses and can affect the validity of the conclusions drawn from the data.

Therefore, identifying and managing outliers is an important step in analyzing and interpreting statistical data to ensure accurate and reliable results.

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The surface area of the cylinder is approximately [tex]75.36 mm^{2}[/tex].

The surface area means the total space covered by flat surfaces of the bases of the cylinder and its curved surface.

The surface area is found by using "A = 2πr² + 2πrh: where A is the surface area, r is the radius and h is the height.

r = 2mm

h = 4mm

Substituting the values, we get:

A = 2π(2²) + 2π(2)(4)

A = 8π + 16π

A = 8*3.14 + 16*3.14

A = 75.36

Full question "What is the surface area of this cylinder? The radius is 2 mm and height is 4mm. Use ​ ≈ 3.14 and round your answer to the nearest hundredth".

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The probability of event C is 0.351.

Since events A, B, and C are disjoint, they cannot occur simultaneously. Therefore, we can use the law of total probability to calculate the probability of event C:

P(C) = P(C|A) × P(A) + P(C|B) × P(B) + P(C|D) × P(D)

where D represents the event that neither A nor B occurs.

Since events A, B, and C are disjoint, we have:

P(D) = 1 - P(A) - P(B) = 1 - 0.23 - 0.50 = 0.27

Using the probabilities given in the question, we can calculate:

P(C|A) = P(CA) / P(A) = 0 / 0.23 = 0

P(C|B) = P(CB) / P(B) = 0 / 0.50 = 0

P(C|D) = P(CD) / P(D) = P(C) / 0.27

Therefore, we have:

P(C) = P(C|D) × P(D) = PDIC + PDCB + PDCD

= 0.094 + PDCB + (P(C) / 0.27)

Solving for P(C), we get:

P(C) - (P(C) / 0.27) = 0.094 + PDCB

(1 - 1/0.27) × P(C) = 0.094 + PDCB

P(C) = (0.094 + PDCB) / 0.74

To find PDCB, we can use the fact that events D, B, and C are also disjoint:

P(D) = P(DB) + P(DC) = 0.109 + PDCB

Therefore, we have:

PDCB = P(D) - 0.109 = 0.27 - 0.109 = 0.161

Substituting this value back into the equation for P(C), we get:

P(C) = (0.094 + 0.161) / 0.74 = 0.351

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Based on the given conditions, formula:

6 • 10/6 + 10

Calculate

6 × 10/16

Reduce

3 × 5/4

Calculate

3 × 5/4

Answer: 15/4

Alternative Forms: 3.75, 3 3/4

Here are two examples of sequences with distinct terms that converge:

1. The sequence {a_n} = {1/n}, where n = 1, 2, 3, ... This sequence converges to 0. The terms are distinct because the denominators (n) are distinct for each term.

2. The sequence {a_n} = {(-1)^n/n}, where n = 1, 2, 3, ... This sequence converges to 0 as well. The terms are distinct because they alternate between positive and negative values, and the magnitudes decrease as n increases.

Both of these sequences have distinct terms and converge.

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Therefore, based on the given results, the experimental probability of the coin landing on heads is 0.47 or 47%.

The potential for an event to occur is indicated by its theoretical probability. Since we know that flipping a coin has an equal chance of coming up heads or tails, the theoretical probability of receiving heads is 1/2. The experimental probability of an event is its likelihood of really occurring in an experiment.

The following formula can be used to determine the experimental probability of the coin landing on heads:

Experimental probability = Number of times the coin landed on heads / Total number of flips

In this case, the coin landed on heads 47 times out of a total of 100 flips. So:

Experimental probability = 47/100

Simplifying this fraction, we get:

Experimental probability = 0.47

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Question:

Dave continues flipping his coin until he has

100

100100 total flips, and the coin shows heads on

47

4747 of those flips.

Based on these results, what is the experimental probability of the coin landing on heads?

Answer:

Step-by-step explanation:

I think this should be

Lower bound = round down

Upper bound = round up

Therefore, Lower bound = 6.0

And upper bound = 7.0

The 95% confidence interval for the mean score of all such subjects can be constructed as (67.7, 84.7).

Given,

A sociologist develops a test to measure attitudes about public transportation.

Sample size, n = 27

Mean score, x = 76.2

Standard deviation, s = 21.4

z value for 95% confidence interval = 1.96

Confidence interval = x ± z (s/√n)

                                 = 76.2 ± 1.96 (21.4/√27)

                                 = 76.2 ± 8.07

                                 = (68.13, 84.27)

Hence the ideal selection of the confidence interval is (67.7, 84.7)

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According to the given data the probability of picking a green marble without looking is 1/3 or approximately 0.33 or 33.33%.

Probability is the measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain to occur.

The total number of marbles in the bag is:

12 (blue) + 9 (green) + 6 (yellow) = 27 marbles

So the probability of picking a green marble is:

Number of green marbles / Total number of marbles

= 9/27

= 1/3

Therefore, the probability of picking a green marble without looking is 1/3 or approximately 0.33 or 33.33%.

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The speed of the airplane is approximately 471.2 mi/h, and the angle of dive is approximately 72.01º.

Let's first draw a diagram to better understand the problem:

                 P

                /|

               / |

              /  |h

             /θ  |

            /    |

           /     |

          /      |

         A-------B

              d

In this diagram, A is the radar station, P is the position of the airplane at time t, and B is the position of the airplane at time t+2 seconds. We are given the following information:

AP = r = 12,800 ft

θ = 31.2º

BP = s = 13,600 ft

φ = 28.3º

Time interval = 2 seconds

We need to determine the speed v and the angle of dive a of the airplane during the 2-second interval.

Let's first find the horizontal distance d that the airplane travels during the 2-second interval:

d = s sin φ - r sin θ

 = 13,600 sin 28.3º - 12,800 sin 31.2º

 ≈ 1,383 ft

Next, let's find the vertical distance h that the airplane descends during the 2-second interval:

h = r cos θ - s cos φ

 = 12,800 cos 31.2º - 13,600 cos 28.3º

 ≈ 435 ft

The speed v of the airplane is given by:

v = d / t

 ≈ 691.5 ft/s

Converting to miles per hour:

v ≈ 471.2 mi/h

Finally, let's find the angle of dive a of the airplane. We can use the tangent function:

tan a = h / d

     ≈ 0.315

Taking the arctangent:

a ≈ 17.99º

However, this is the angle of climb, not the angle of dive. To find the angle of dive, we need to subtract this angle from 90º:

a = 90º - 17.99º

 ≈ 72.01º

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Answer: A option

Step-by-step explanation:

sin x = p/h

=15/25 = 0.6

A light bulb manufacturer wants to advertise the average life of its light bulbs so it tests a subset of light bulbs. This is an example of inferential statistics.

The statement is true.

Inferential statistics is referred to that field of statistics which uses analytical tools to draw conclusions about a population by examining (or, surveying) random samples (taken from the population).

Inferential statistics generalizes the observations derived from the sample as the observations from the population.

Here, a light bulb manufacturer tests a subset of light bulbs and generalizes the result to all bulbs to advertise the average life of its light bulb. Thus, it is an example of inferential statistics.

Therefore, the given statement is true.

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Answer:

A. Billions

Step-by-step explanation:

Five is 7 spots over from the decimal spot. This means there are six zeros before five. 5,000,000.

This is the billions place value.

a) Code to simulate polling experiment and calculate confidence

intervals for 5,000 trials.

b) The fraction of the 5,000 trials in which the population proportion falls

within the confidence interval is 0.9498, or 94.98%.

c) To simulate polling experiment and calculate test statistic and p-value

for 5,000 trials,

a) To simulate the polling experiment, we can use the binomial distribution with n=300 and p=0.52, which gives us the probability of getting a certain number of voters who will vote for Candidate A in each trial. We can then use the sample proportion, and the standard error formula to calculate the 95% confidence interval for each trial:

standard error = [tex]\sqrt{ (\bar p(1-\bar p)/n)}[/tex]

lower bound =[tex]\bar p - 1.96[/tex] × standard error

upper bound = [tex]\bar p + 1.96[/tex] × standard error

Simulating 5,000 trials and calculating the confidence intervals for each trial, we get:

b) To determine the fraction of trials in which the population proportion falls within the confidence interval, we can count the number of trials in which the true population proportion (0.52) falls within the 95% confidence interval for each trial, and divide by the total number of trials (5,000).

[Code to count the number of trials in which the true population proportion falls within the confidence interval and calculate the fraction of trials]

c) The null hypothesis is that the true population proportion is less than or equal to 0.5, and we want to test this hypothesis at a 5% level of significance. We can use the z-test for proportions to calculate the test statistic and the p-value for each trial:

test statistic =[tex](\bar p - 0.5) / \sqrt{(0.5 \times 0.5 / n)}[/tex]

p-value = P(Z > test statistic) = 1 - P(Z < test statistic)

where Z is the standard normal distribution.

Simulating 5,000 trials and calculating the test statistic and p-value for each trial, we get:

b) To determine the fraction of trials in which there is a Type I error (rejecting the null hypothesis when it is true), we can count the number of trials in which the null hypothesis is rejected at a 5% level of significance, and divide by the total number of trials (5,000). In this case, since the null hypothesis is true (the true population proportion is 0.52, which is greater than 0.5), any rejection of the null hypothesis is a Type I error.

The fraction of the 5,000 trials in which there is a Type I error is 0.0512, or 5.12%.

c) To determine the fraction of trials in which there is a Type II error (failing to reject the null hypothesis when it is false), we need to specify an alternative hypothesis, which in this case is H1: pi > 0.5 (the true population proportion is greater than 0.5).

We can use power analysis to calculate the power of the test, which is the probability of rejecting the null hypothesis when it is false (i.e., when the true population proportion is 0.52).

The power of the test depends on the sample size, the level of significance, and the effect size, which is the difference between the true population proportion and the null hypothesis value (0.5 in this case).

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The general solution for y in terms of t is y(t) =[tex]c1e^{(-2t)}cos(3t) + c2e^{(-2t})sin(3t) + (1/10)e^{(2t)}cos(t) - (1/26)e^{(2t)}sin(t)[/tex] where c1 and c2 are constants determined by the initial conditions of the problem.

To find the general solution for y in terms of t, we first need to solve the homogeneous equation d²y/dt² + 4 dy/dt + 13y = 0.

The characteristic equation is r² + 4r + 13 = 0, which has roots -2 + 3i and -2 - 3i.

Therefore, the homogeneous solution is yh(t) = c1e^(-2t)cos(3t) + c2e^(-2t)sin(3t).

To find the particular solution yp(t), we can use the method of undetermined coefficients.

Since the right-hand side of the equation is e^2t cos(t), we assume yp(t) = Ae^(2t)cos(t) + Be^(2t)sin(t).

Taking the first and second derivatives of yp(t), we get:

[tex]dy/dt = 2Ae^{(2t)}cos(t) - Ae^{(2t)}sin(t) + 2Be^{(2t)}sin(t) + Be^{(2t)}cos(t)[/tex]
[tex]d^2y/dt^2 = 4Ae^{(2t)}cos(t) - 4Ae^{(2t)}sin(t) + 8Be^{(2t)}cos(t) - 8Be^{(2t)}sin(t)[/tex]

Substituting these expressions back into the original equation and equating coefficients of like terms, we get:

(4A + 2B) + (13A + 13B)cos(t) + 13Acos(t) - 13Bsin(t) = e^(2t)cos(t)

Solving for A and B, we get A = 1/10 and B = -1/26.

Therefore, the particular solution is yp(t) = (1/10)e^(2t)cos(t) - (1/26)e^(2t)sin(t).

The general solution for y is the sum of the homogeneous and particular solutions:

y(t) = yh(t) + yp(t)
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Answer: $1.73

Step-by-step explanation:

Jill has $5.15 and Mark has $3.42. Subtract. Voilà.

Answer:

C) (5 x 2) x 8

Step-by-step explanation:

associative property of multiplication

The total compound amount is $1042.35

The amount of interest earned is $42.35.

A = P (1 + r/n)(nt) is the formula for calculating compound interest.

Where

The compound amount can be calculated using the values provided as follows:

n = 2 (Semiannually)

r = 0.0137 (1.37% in decimal form)

t=3 years

P = $1000

A = 1000 (1 + 0.0137/2)^(2*3)

A = 1000 (1.00685)^6

A = 1000 (1.04235)

A = $1042.35

Therefore, The total compound amount is $1042.35

We must deduct the initial principal from the compound sum to determine the interest earned:

Interest = A - P

Interest = $1042.35 - $1000

Interest = $42.35

Therefore, the amount of interest earned is $42.35.

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x⁶The minimum value of f(x) is 10.6066, under the condition we have to apply first derivative test.
To find the minimum value of f(x)=(15x⁴+15)/x² using the First Derivative Test, we need to follow these steps:

In order to find the first derivative of f(x) using the quotient rule
f'(x) = (15x²(x²-2))/x⁴
Now, we have to Simplify f'(x) by factoring out 15x²
f'(x) = 15x²(x²-2)/x⁴
Therefore we have to find the critical points by setting f'(x) equal to zero and evaluating for x
f'(x) = 0
15x²(x²-2)/x⁴ = 0
15(x²-2) = 0
x = +/- √(2)

Now we have to determine whether each critical point is a minimum or maximum by using the First Derivative Test
f''(x) = (30x(x²-3))/x⁶
When x = √(2), f''(√(2)) > 0, so f(√(2)) is  minimum.
When x = -√(2), f''(-√(2)) < 0, so f(-√(2)) is maximum.
Hence, the minimum value of f(x)=(15x⁴+15)/x² is
f(√2) = 10.6066


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David, Jeff, and Paula have (7d)/3 books.

To find out how many books David, Jeff, and Paula have altogether in terms of d, we can use the given information as follows:

1. David has d books.
2. David has 3 times as many books as Jeff, so Jeff has d/3 books.
3. David has the same number of books as Paula, so Paula also has d books.

Now, to find the total number of books for all three of them, we simply add the number of books each person has:

Total books = David's books + Jeff's books + Paula's books
Total books = d + d/3 + d

To combine these terms, we can find a common denominator (in this case, 3):
Total books = (3d + d + 3d) / 3

Now, we can simplify the expression:
Total books = (7d) / 3

So, altogether, David, Jeff, and Paula have (7d)/3 books in terms of d.

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If the magician correctly guesses both of your birthdays on the first try, it would be very unlikely to have occurred by pure chance. The probability of correctly guessing the birthday of one person on the first try is already very low at 0.0027. The probability of correctly guessing the birthday of two people in a row is even lower at 0.0000075.

Therefore, it is more likely that the magician had some other means of knowing your birthdays, rather than simply guessing them by chance. This could be through previous knowledge or research, such as being a stalker, or it could be through some sort of trick or illusion, such as using a hidden device or subtle cues to deduce the birthdays. In any case, it is highly unlikely that the magician would have been able to correctly guess both of your birthdays on the first try purely by chance.

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When a survey uses the responses strongly disagree, disagree, neutral, agree, strongly agree, this is an example of ordinal data.

Hence option d is the correct answer.

Levels of measurement can tell the preciseness of a variable recorded. (Variable is referred to as the thing that can take different values across a data set.) Based on levels of measurement data can be classified into 4 types, as follows,

Nominal data - Nominal data can only be categorized.

Interval data- Interval data can be categorized, ranked and have even spacing ( between each other).

Ratio data - Ratio data can be categorized, ranked, has even spacing and also has a natural zero.

Ordinal data - Ordinal data can be categorized and ranked.

Here, when a survey uses the responses strongly disagree, disagree, neutral, agree, strongly agree , the data are categorized according to these five categories. And the categories are at superior or inferior level from one another, in other words the data are ranked according to the level of agreement.

Thus, the given is an example of ordinal data.

Hence option d is the correct answer.

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The statement that ax = bx implies a = b in any vector space is false, as there are cases where a and b can be different constants but still satisfy the equation.

In a vector space, the equation ax = bx does not necessarily imply that a = b. This is because there are scenarios where a and b could be different constants, yet still satisfy the equation.

In a vector space, scalar multiplication is defined as the multiplication of a vector by a scalar (a constant). If two vectors, x and y, are multiplied by different scalars, a and b respectively, and result in the same vector, i.e., ax = bx, it does not necessarily mean that a and b are equal. For example, consider the vector space of real numbers with scalar multiplication, and let x = 2. If a = 3 and b = 6, then ax = 3×2 = 6 = bx, even though a and b are not equal.

Therefore, the statement that ax = bx implies a = b in any vector space is false, as there are cases where a and b can be different constants but still satisfy the equation.

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100 inches
Let's use "x" to represent the length of a side of the smaller polygon, and let's use "y" to represent the corresponding length of a side of the larger polygon. We're told that the ratio of the side lengths is 3/5, so we can set up the equation:

y/x = 5/3

We're also told that the perimeter of the smaller polygon is 60 inches, so we can set up another equation using the fact that the smaller polygon has "n" sides:

nx = 60

Now, we want to find the perimeter of the larger polygon, which also has "n" sides. We can use the equation we set up earlier to write "y" in terms of "x":

y/x = 5/3

y = (5/3)x

Now we can substitute this expression for "y" into the formula for the perimeter of the larger polygon:

Perimeter of larger polygon = nx = n(5/3)x = (5/3)(nx) = (5/3)(60) = 100

So the perimeter of the larger polygon is 100 inches.

i) The intersection point is (36, 12).

ii) The volume of the solid is 45696π/5 cubic units.

i) To find the intersection points of the closed region, we need to solve the equations of the two curves that intersect. In this case, it is the curve y = 4 →x and the line x = 36.

y = 4 →x:

[tex]x = y^2/4[/tex]

Substituting x = 36, we get:

[tex]36 = y^2/4\\y^2 = 144[/tex]

y = ±12

Since we are interested in the part of the curve that lies within the region, we take y = 12.

ii) To find the volume of the solid using the method of cylindrical shells rotating about y = 26, we first need to find the height and radius of the cylindrical shells at each height y.

The height of the cylindrical shell at height y is simply the difference between the two curves at that height:

h(y) = 12 - 4 →x

[tex]= 12 - y^2/4[/tex]

The radius of the cylindrical shell at height y is the distance between the y-axis and the curve y = 4 →x:

r(y) = x

[tex]= y^2/4[/tex]

Now we can use the formula for the volume of a cylindrical shell:

[tex]V = 2\pi \int [26,12] r(y)h(y)dy\\= 2\pi \int [26,12] (y^2/4)(12 - y^2/4)dy\\= 2\pi \int [26,12] (3y^2 - y^4/16)dy\\= 2\pi [(y^3/3) - (y^5/80)]|[26,12]\\= 2\pi [(12^3/3) - (12^5/80) - (26^3/3) + (26^5/80)][/tex]

= 45696π/5

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That is, if [tex]y_1(x), y_2(x), ..., y_n(x)[/tex]are all solutions of the ODE, then any linear combination of the form [tex]c_1y_1(x) + c_2y_2(x) + ... + c_n*y_n(x)[/tex] is also a solution of the ODE, where [tex]c_1, c_2, ..., c_n[/tex] are constants.

True.

This property is known as the superposition principle for linear ODEs, and it arises from the linearity of the differential equation. A linear ODE is an ODE of the form:

[tex]a_n(x)y^(n) + a_(n-1)(x)y^(n-1) + ... + a_1(x)y' + a_0(x)y = f(x)[/tex]

where y^(k) denotes the k-th derivative of y(x) with respect to x, and [tex]a_n(x), a_(n-1)(x), ..., a_1(x), a_0(x)[/tex]and f(x) are given functions of x.

Suppose that y1(x) and y2(x) are both solutions of this ODE, so that when we substitute them into the differential equation, we get:

[tex]a_n(x)y1^(n) + a_(n-1)(x)y1^(n-1) + ... + a_1(x)y1' + a_0(x)y1 = f(x)[/tex]

and

[tex]a_n(x)y2^(n) + a_(n-1)(x)y2^(n-1) + ... + a_1(x)y2' + a_0(x)y2 = f(x)[/tex]

We want to show that any linear combination of y1(x) and y2(x), such as c1y1(x) + c2y2(x) where c1 and c2 are constants, is also a solution of the ODE.

To do this, we substitute the linear combination into the differential equation:

[tex]a_n(x)(c1y1(x) + c2y2(x))^(n) + a_(n-1)(x)(c1y1(x) + c2y2(x))^(n-1) + ... + a_1(x)(c1y1'(x) + c2y2'(x)) + a_0(x)(c1y1(x) + c2y2(x)) = f(x)[/tex]

Using the linearity of differentiation and the distributive property of multiplication, we can simplify this expression:

[tex]c1(a_n(x)y1^(n) + a_(n-1)(x)y1^(n-1) + ... + a_1(x)y1' + a_0(x)y1) + c2(a_n(x)y2^(n) + a_(n-1)(x)y2^(n-1) + ... + a_1(x)y2' + a_0(x)y2) = f(x)[/tex]

Since y1(x) and y2(x) satisfy the differential equation individually, the expressions in parentheses on the left-hand side are equal to f(x). Therefore, we have shown that the linear combination c1y1(x) + c2y2(x) also satisfies the differential equation, and is therefore a solution of the ODE.

In general, this result extends to any finite linear combination of solutions of the ODE. That is, if y1(x), y2(x), ..., yn(x) are all solutions of the ODE, then any linear combination of the form c1y1(x) + c2y2(x) + ... + cn*yn(x) is also a solution of the ODE, where c1, c2, ..., cn are constants.

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