The first eight nonzero terms of the solution by substituting the values of the coefficients in the power series expression for y(x) are:
[tex]a_0 = 0\\a_1 = -a_2/2\\a_2 = -3a_1/2\\a_3 = -(4a_2 + 6a_1)/2\\a_4 = -(5a_3 + 7a_2)/2\\a_5 = -(6a_4 + 8a_3)/2\\a_6 = -(7a_5 + 9a_4)/2\\a_7 = -(8a_6 + 10a_5)/2\\a_8 = -(9a_7 + 11a_6)/2[/tex]
To solve the given differential equation y" + xy' + 2y = 0 using power series method around xo = 0, we assume a power series solution of the form:
y(x) = ∑[n=0 to ∞] [tex]a_n x^n[/tex]
Now, let's find the first eight nonzero terms of this power series solution.
First, we'll calculate the derivatives of y(x) with respect to x:
y'(x) = ∑[n=0 to ∞] a_n * n * [tex]x^{(n-1)[/tex]
y''(x) = ∑[n=0 to ∞] a_n * n * (n-1) * [tex]x^{(n-2)[/tex]
Substituting these expressions into the original differential equation, we have:
∑[n=0 to ∞] [tex]a_n[/tex] * n * (n-1) * [tex]x^{(n-2)[/tex] + x * ∑[n=0 to ∞] [tex]a_n[/tex] * n * [tex]x^{(n-1)[/tex] + 2 * ∑[n=0 to ∞] [tex]a_n * x^n[/tex] = 0
Now, we'll rearrange the terms and combine them:
∑[n=2 to ∞][tex]a_n[/tex] * n * (n-1) * [tex]x^{(n-2)[/tex] + ∑[n=1 to ∞] [tex]a_n[/tex] * n * [tex]x^n[/tex] + 2 * ∑[n=0 to ∞] [tex]a_n * x^n[/tex] = 0
Let's break down each summation separately:
For the first summation term, n starts from 2:
∑[n=2 to ∞] [tex]a_n[/tex] * n * (n-1) * [tex]x^{(n-2)[/tex] = a_2 * 2 * 1 * [tex]x^0[/tex] + [tex]a_3[/tex] * 3 * 2 * x^1 + [tex]a_4[/tex] * 4 * 3 * [tex]x^2[/tex] + ...
For the second summation term, n starts from 1:
∑[n=1 to ∞] [tex]a_n[/tex] * n * [tex]x^n[/tex] = [tex]a_1[/tex] * 1 * [tex]x^1[/tex] + [tex]a_2[/tex] * 2 * [tex]x^2[/tex] + [tex]a_3[/tex] * 3 * [tex]x^3[/tex] + ...
For the third summation term, n starts from 0:
2 * ∑[n=0 to ∞] [tex]a_n * x^n[/tex] = 2 * [tex]a_0 * x^0[/tex] + 2 * [tex]a_1 * x^1[/tex] + 2 *[tex]a_2 * x^2[/tex] + ...
Combining these terms, we have:
2[tex]a_0[/tex] + (2[tex]a_1 + a_2[/tex])x + (2[tex]a_2 + 3a_1[/tex])[tex]x^2[/tex] + [tex](2a_3 + 4a_2 + 6a_1)x^3[/tex] + ...
Since the equation should hold for all values of x, each coefficient of [tex]x^n[/tex]should be zero. Therefore, we equate each coefficient to zero and find the recurrence relation for the coefficients:
2[tex]a_0[/tex] = 0 => [tex]a_0 = 0[/tex]
[tex]2a_1 + a_2 = 0[/tex] => [tex]a_1 = -a_2/2[/tex]
[tex]2a_2 + 3a_1 = 0[/tex] => [tex]a_2 = -3a_1/2[/tex]
[tex]2a_3 + 4a_2 + 6a_1 = 0[/tex]
Using these recurrence relations, we can calculate the first eight nonzero terms of the solution.
Starting from [tex]a_0 = 0[/tex], we can find the values of [tex]a_1, a_2[/tex], and so on:
[tex]a_0 = 0[/tex]
[tex]a_1 = -a_2/2[/tex]
[tex]a_2 = -3a_1/2[/tex]
[tex]a_3 = -(4a_2 + 6a_1)/2[/tex]
[tex]a_4 = -(5a_3 + 7a_2)/2[/tex]
[tex]a_5 = -(6a_4 + 8a_3)/2[/tex]
[tex]a_6 = -(7a_5 + 9a_4)/2[/tex]
[tex]a_7 = -(8a_6 + 10a_5)/2[/tex]
[tex]a_8 = -(9a_7 + 11a_6)/2[/tex]
Therefore, these are the first eight nonzero terms of the solution by substituting the values of the coefficients in the power series expression for y(x).
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Question Progress
boel
21/40 Marks
Find the area of this parallelogram.
13 cm
15 cm
20 cm
Picking a purple marble from a jar with 3 green and 3 purple marbles. What is the probability of picking a purple marble?
Answer:
1/2
Step-by-step explanation:
The total number of marbles in the jar is equal to 6
Since we have 3 purple marbles and 3 green marbles,
The probability of picking a purple marble =
Number of purple marbles in the jar / total number of marbles
= 3/6
= 1/2
From this calculation the probability of picking a purple marble is 1/2 or 0.5
1. What is 5,509 rounded to the thousands place?
А
6,000
B
5,500
5,600
D
5,000
X Kawika had to raise the flag this morning. The 15 ft flag pole casted at
shadow with the ground at a 35° angle. What is the length of the
shadow? (Round answers to the nearest tenth) *
Answer with JUST THE NUMBER, without writing out the unit after the number.
please please I really need help this depends on my life
Answer:
39.4 divided by (-7.2)= -5.472
6.7-39.31=-32.61
Step-by-step explanation:
Help there are five people at the carnival they are 15 people at the other carnival how many people are there
The diameter of a circle is 4 feet. What is the circumference?
d=4 ft
Give the exact answer in simplest form.
I need the answers
Answer: 4π
Step-by-step explanation: C= 2πr; 2r=d; C=dπ; C=4π!
Answer:
4π ft
Step-by-step explanation:
C = 2πr (Circumference formula for a circle)
diamater = 4, so diamter = 2 x radius, 4 = 2 x r , so r = 2
Use formula:
C = 2π(2) = 4π
4π ft (Exact answer)
Minh wants to buy a car costing $9,950. She can put $1,000 down and finance the rest at 10.9% APR, making the monthly payments for 3 years. How will Minh pay in finance charges? What will her monthly payments be?
Answer:
piip;[-p-o-ul0i
Step-by-step explanation:
Is x = 5 a solution to the equation 19 - x = 14?
Answer:
Yes
Step-by-step explanation:
Test the equation by substituting 5 in for x:
19- 5 =14
14=14
TRUE
If I helped, a brainliest would be greatly appreciated!
Answer: If you replace x with 5 then 19-5=14
Step-by-step explanation:
Please help! if do you will get 'Brainliest' (Right answers only, and please show your work)
Answer:
The answer should be B
[tex]3c \: + \: 43 \: \geqslant \: 100[/tex]
Help me please with the answer
tell me how to answer this
The mid point of the line segment is (-2, 5 / 2).
How to find the mid point of a line segment?The midpoint of a line segment is a point that lies exactly halfway between two points.
Therefore, let's find the mid point of the line segment indicated above.
Hence, using (-4, 0) and (0, 5)
Hence,
mid point = (x₁ + x₂ / 2, y₁ + y₂ / 2)
Therefore,
mid point = (0 - 4 / 2, 5 + 0 / 2)
mid point = (-4 /2, 5 / 2)
mid point = (-2, 5 / 2)
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24 Hour Fitness charges svara spatee plus so per months to poun thensyn time has podszes por s gym membership so farHow many monohis has he been a member at 24 Hour Fitness monks
Answer:
4 months
Step-by-step explanation:
Equation:
y = 55 + 30x
y = gym membership
x = per month
Work:
y = 55 + 30x
175 = 55 + 30x
30x = 175 - 55
30x = 120
x = 4
4 months
7 + k = 11 step explanation
Answer:
k = 4
Step-by-step explanation:
11 - 7 = 4
Using mirrors installed on the moon by Apollo astronauts to reflect laser waveforms, its possible to measure the distance from earth to moon to high precision. Due to the physics of the detectors, the measurement errors have a uniform distribution of -3.0 to + 5.0 cm
a. What is the probability that a particle measurement will be accurate to within +- 1.0 cm?
b. Find the mean of the measurement errors
c. Find the standard deviation of the measurement errors
A.the probability that a particle measurement will be accurate to within +- 1.0 cm is 0.25 B. the mean of the measurement errors is 1 cm. C. the standard deviation of the measurement errors is 2.311 cm.
a. For a uniform distribution of -3.0 to + 5.0 cm, the total distance is 5.0 - (-3.0) = 8.0 cm.
The probability that a particle measurement will be accurate to within +- 1.0 cm is given by:
P(-1.0 ≤ X ≤ 1.0) = (1/(8.0)) * (1.0 - (-1.0))= (1/8.0) * 2= 0.25
b. Find the mean of the measurement errors
The formula to calculate the mean (μ) of the measurement errors is:μ = (a + b) / 2
where, a is the lower limit of the distribution, and b is the upper limit of the distribution.μ = (-3.0 + 5.0) / 2= 2 / 2= 1
Therefore, the mean of the measurement errors is 1 cm.
c. Find the standard deviation of the measurement errors
The formula to calculate the standard deviation (σ) of a uniform distribution is:σ = (b - a) / √12
where, a is the lower limit of the distribution, b is the upper limit of the distribution, and √12 is the square root of 12.σ = (5.0 - (-3.0)) / √12= 8 / 3.464= 2.311
Therefore, the standard deviation of the measurement errors is 2.311 cm.
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PLEASE HELP PLEASE
Write a clear explanation summarizing what you have learned about defining expressions that have zero or a negative integer as an exponent. Then explain, using examples, why these definitions makes sense. Give as many reasons as you can. Also indicate which exclamation makes the most sense to you.
Answer:
Any number raised to zero is one.
Examples:
[tex]4^{0} = 1\\\\178^0=1\\\\15^0 = 1[/tex]
Any number raised to a negative integer is 1 over that exponential term without the negative exponent.
Examples:
[tex]5^{-1} = \frac{1}{5}\\\\324^{-13}=\frac{1}{324^{13}}\\\\17^{-6} = \frac{1}{17^{6}}[/tex]
Someone please help me.
Answer:
with what life
Step-by-step explanation:
what is the first 5 cubic numbers
Answer:
1,8,27,64,125
Step-by-step explanation:
Please help.
a+67a+3ab-2ab-1=
Answer:
68a+ab-1
Step-by-step explanation:
Have a good day!
"
Perform the following test of hypothesis. H0: μ 10.70, H1: μ # 10.70, n = 47, t = 12.234, = H0 is
"
Based on the given information, we have sufficient evidence to conclude that the population mean (μ) is not equal to 10.70.
To perform the test of hypothesis, we can use a t-test to determine whether there is sufficient evidence to reject the null hypothesis (H0) in favor of the alternative hypothesis (H1).
The given hypotheses are:
H0: μ = 10.70 (null hypothesis)
H1: μ ≠ 10.70 (alternative hypothesis)
The sample size is n = 47, and the test statistic is t = 12.234.
To conduct the test, we need to compare the calculated test statistic to the critical value(s) based on the desired significance level (α).
The steps to perform the hypothesis test are as follows:
Step 1: Set the significance level (α)
Let's assume a significance level of α = 0.05. This means we want to be 95% confident in our decision.
Step 2: Determine the critical value(s)
Since the alternative hypothesis is two-sided (μ ≠ 10.70), we need to find the critical values for a two-tailed t-test.
Since n = 47, we have n - 1 = 46 degrees of freedom (df). Looking up the critical values in the t-distribution table or using a statistical calculator, the critical values for a two-tailed test with α = 0.05 and df = 46 are approximately ±2.013.
Step 3: Calculate the p-value
The p-value is the probability of obtaining a test statistic as extreme as the one observed (t = 12.234), assuming the null hypothesis is true.
Using a t-table or statistical software, we can determine the p-value associated with the given t-value and degrees of freedom. The p-value turns out to be very small (p < 0.001).
Step 4: Make a decision
If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Since the p-value is smaller than α (p < 0.001), we reject the null hypothesis (H0).
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write the equation of a function whose parent function, f(x) = x 8, is shifted 2 units to the right.
To find the equation of a function whose parent function, f(x) = x^8, is shifted 2 units to the right, we can use the formula given below: g(x) = f(x - a) + bwhere f(x) is the parent function, a is the horizontal shift (positive for right and negative for left), b is the vertical shift (positive for up and negative for down), and g(x) is the new function.So, in this case, we need to shift the parent function f(x) = x^8 two units to the right.
This means that a = 2 and b = 0 (since there is no vertical shift).So, substituting these values in the formula, we get:g(x) = f(x - 2) + 0g(x) = (x - 2)^8Thus, the equation of the new function, g(x), is (x - 2)^8 when the parent function, f(x) = x^8, is shifted 2 units to the right.
To shift the parent function f(x) = x^8 two units to the right, we need to modify the equation by replacing 'x' with 'x - 2'. This will shift the graph horizontally to the right by 2 units.
Therefore, the equation of the shifted function, let's call it g(x), becomes:
g(x) = (x - 2)^8
To shift the parent function, f(x) = x^8, two units to the right, we can use the transformation equation:
g(x) = f(x - h)
where h represents the horizontal shift. In this case, h = 2 because we want to shift the function two units to the right. Substituting the values into the equation, we get:
g(x) = (x - 2)^8
Therefore, the equation of the function after shifting the parent function, f(x) = x^8, two units to the right is g(x) = (x - 2)^8.
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The given parent function is f(x) = x 8. We are supposed to write the equation of a function that is shifted two units to the right. It can be done by taking a new function g(x) and writing it as g(x) = f(x − 2).
Therefore, the equation of the new function g(x) is g(x) = (x − 2)8.
Step-by-step explanation: The given parent function is f(x) = x 8. We are supposed to write the equation of a function that is shifted two units to the right. It can be done by taking a new function g(x) and writing it as g(x) = f(x − 2).
The value of f(x) = x 8
Write it as follows:
when x is 0, 0 8 = 0 and
when x is 1, 1 8 = 8.
Now, we can write the equation for g(x) by using f(x) = x 8.
We have g(x) = f(x − 2)
= (x − 2)8.
Therefore, the equation of the new function g(x) is g(x) = (x − 2)8.
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For a hypothesis test using the single-factor ANOVA formula:
The obtained overall F ratio = MS between treatments divided by MS within treatments.
Calculate the obtained overall F ratio for: MS between treatments= 8 and MS within treatments = 2.
Then describe the null hypothesis decision for = .05, df numerator: 3, and df denominator: 4.
Reject of fail to reject null hypothesis?
We can conclude that there is not enough evidence to suggest that the means of the different groups are different.
The obtained overall F ratio for MS between treatments = 8 and MS within treatments = 2 can be calculated using the formula for a hypothesis test using the single-factor ANOVA. The formula is given below: Obtained overall F ratio = MS between treatments/MS within treatments Substituting the values in the above formula, we get, Obtained overall F ratio = 8/2 = 4
The obtained overall F ratio for MS between treatments = 8 and MS within treatments = 2 is 4.Now, let's describe the null hypothesis decision for α = .05, df numerator: 3, and df denominator: 4.The null hypothesis decision can be made using the F-distribution table. Using the table, the critical value of F is 9.49.
Therefore, if the obtained overall F ratio is greater than 9.49, then we reject the null hypothesis. If the obtained overall F ratio is less than or equal to 9.49, then we fail to reject the null hypothesis.Obtained overall F ratio = 4 is less than the critical value of F, i.e., 9.49.Therefore, the null hypothesis is not rejected at α = .05, df numerator: 3, and df denominator: 4.
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The obtained overall F ratio for MS between treatments = 8 and MS within treatments = 2 will be 4. The null hypothesis decision for α = 0.05, df numerator: 3, and df denominator: 4 is Fail to reject null hypothesis.
Explanation: ANOVA (analysis of variance) is a statistical method used to determine if two or more population means are equal.
It is used to analyze the variation between sample group means and is frequently used in hypothesis testing.
Therefore, we will now calculate the obtained overall F ratio for the given values.
The obtained overall F ratio formula is: Overall F Ratio = MS between treatments / MS within treatments.
Given, MS between treatments = 8 and MS within treatments = 2.
Therefore, Overall F Ratio = MS between treatments / MS within treatments
= 8/2
= 4
Hence, the obtained overall F ratio for MS between treatments= 8 and MS within treatments = 2 is 4.
The null hypothesis decision for α = 0.05, df numerator: 3, and df denominator: 4 is Fail to reject null hypothesis.
The test statistic value, F, is 3.71. The p-value is greater than 0.05. Therefore, we fail to reject the null hypothesis.
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Emily is making a flag with the dimensions shown. What is the minimum amount of material needed for the flag?
9514 1404 393
Answer:
B 80 square inches
Step-by-step explanation:
The area of a parallelogram has the same formula as the area of a rectangle:
A = bh . . . . . b is the base; h is the height
A = (10 in)(8 in)
A = 80 in²
The minimum amount of material needed is 80 square inches.
Answer:
The answer is 80
Step-by-step explanation:
Hope this helps
pls help i’m so confused omg!!
Answer:
C or B
Step-by-step explanation:
Answer:
Your answer is A.
Step-by-step explanation:
You can count the arrow lines as the 2s. Such as, 2, 4, 6, 8 and 10.
So, the line 'x' is being divided by how many lines touching it.
Therefore, it is 1.
Then you add x and 6, the number which the line is ending with.
Your final answer is, F(x) = 1
x+6
Hope this helped!
The circle graph shows how William spends his day on average. If he sticks
to this schedule, how many hours will William spend sleeping over the
course of a week (7 days)? ()
17% 11%
10%
А
A) 75.6 hours
B) 78.2 hours
C) 814 hours
Watching TV
Video Games
Fating
Working
45 %Sleeping
27%
D) 848 hours
Answer:
d
Step-by-step explanation:
i did the math
Multiplication and division questions that equal -8?
And another one for -15?
Pls and thank u <3
For negative 8
1. -4 x 2
2. -48 ÷ 8
For negative 15
1. -5 x 3
2. -60 ÷ 4
( Sorry if they aren't complex enough)
Solve for n. 2n+3/4n = 3/4
Answer: n=3/11
Step-by-step explanation:
1. You want to simplify both sides of the equation.
2n+3/4n=3/4.
(2n+3/4n)=3/4 (combine like terms).
11/4n=3/4.
2. Multiply both sides using 4/11.
(4/11)x(11/4n)=(4/11)x(3/4)
3. You get your answer which is n=3/11.
Hoped I help :)
The center and a point on a circle are given. Find the circumference to the nearest tenth.
center:(−15, −21);
point on the circle: (0, −13)
The circumference is about ?
Answer:
Circumference ~ 106.8
Step-by-step explanation:
Use the distance formula to find the radius of the circle with the two coordinates given. The radius of this circle is 17. Plug 17 into the formula 2(pi)r to find the Circumference. The answer is 106.8 rounded to the nearest tenth.
Tony borrows $7,000 from a friend
at a rate of 5% interest for 4 months.
What is the total that needs
to be repaid after this time
to the nearest dollar?
Answer:
5%100 ×R200
Step-by-step explanation:
Address and u the one for me please
What's the equation to use to solve x?
2222222 help me plz plz plz plz
Answer:
○ B) m∠XOA = m∠OYC
Step-by-step explanation:
Since Point O is circumcentered into the square, we keep it in the midst when we are making comparisons, and since this was simple to identify, this automatically is a false statement. Point O stays in the midst at all times.
- Hope this helps!