To interpolate the function f(x) = x ln(2+1) by a second-order polynomial on equidistant nodes on the interval (0,1), we can use the Lagrange interpolation formula.
To interpolate f(T) by a second-order polynomial on equidistant nodes on (0,1), we need three equidistant nodes. Let's denote these nodes as x₀, x₁, and x₂, with x₀ = 0, x₁ = h, and x₂ = 2h, where h = (1-0)/2 = 1/2.
We can construct the Lagrange polynomial P₂(x) of degree 2 that interpolates f(x) at these nodes. The Lagrange interpolation formula for P₂(x) is:
P₂(x) = f(x₀) × L₀(x) + f(x₁) × L₁(x) + f(x₂) × L₂(x)
where L₀(x), L₁(x), and L₂(x) are the Lagrange basis polynomials.
Using the given function f(x) = x ln(2+1), we can evaluate f(x₀), f(x₁), and f(x₂). Plugging these values into the Lagrange interpolation formula, we obtain the second-order polynomial interpolation for f(T).
To estimate the error, we can use the error formula for polynomial interpolation. The error term E(x) is given by:
E(x) = f(x) - P₂(x)
To calculate the error, we would need to evaluate the derivative of f(x) and find its maximum value on the interval (0,1). However, without knowing the interval (T), we cannot estimate the error.
In summary, we can interpolate f(T) by a second-order polynomial on equidistant nodes on (0,1) using the Lagrange interpolation formula. However, without specifying the interval (T), it is not possible to estimate the error.
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In a fruit cocktail, for every 30ml of orange juice you need 20ml of apple juice and 50ml of coconut milk. What proportion of the cocktail is orange juice? Give your answer in the simplest form of ratio
Answer is in a photo. I can only upload it to a file hosting service. link below!
bit.[tex]^{}[/tex]ly/3a8Nt8n
Answer:
me
Step-by-step explanation:
becssu imthe best guy
Which expressions are equivalent to the one below? Check all that apply 5^x
Answer:
5 * 5^(x - 1) ; (15/3)^x ; 15^x / 3^x
Step-by-step explanation:
From the options, equivalent expressions include :
(15/3)^x
This is the same as ;
(15/3)^x
15 ÷ 3 = 5 ; then to the power of x = 5^x
15^x / 3^x ; since they are both raised to the same power, we can divide directly to obtain :
5^x
5 * 5^(x - 1)
5 = 5^1
5^1 * 5^(x-1)
5^(1 + x - 1) = 5^x
what is the volume of each cylinder with a radius of 2.7 cm and a height of 5 cm
Answer:
114.51
Step-by-step explanation:
I'm not to sure what you meant by 'each' so I solved it like there was only one cylinder. hope this helped
One catalog offers a jogging suit in two colors, gray and black. It comes in sizes S, M, L, XL and XXL. How many possible jogging suits can be ordered? PLEASE HELP NO LINKS!!!
Answer:
5..
Step-by-step explanation:
To test the hypothesis that the population standard deviation sigma-7.2, a sample size n=7 yields a sample standard deviation 5.985. Calculate the P- value and choose the correct conclusion. Your answer: The P-value 0.343 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.343 is The P-value 0.343 is significant and so strongly suggests that sigma<7.2. The P-value 0.192 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.192 is significant and so strongly suggests that sigma<7.2. The P-value 0.291 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.291 is significant and so strongly suggests that sigma<7.2. suggests that sigma<7.2. The P-value 0.309 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.309 is significant and so strongly suggests that sigma<7.2. The P-value 0.011 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.011 is significant and so strongly suggests that sigma<7.2.
The P-value of 0.343 is not significant and does not strongly suggest that the population standard deviation, sigma, is less than 7.2.
In hypothesis testing, the P-value is used to determine the strength of evidence against the null hypothesis. In this case, the null hypothesis is that the population standard deviation, sigma, is equal to 7.2. The alternative hypothesis is that sigma is less than 7.2.
To calculate the P-value, we need to compare the sample standard deviation, which is 5.985, to the hypothesized population standard deviation of 7.2. We can use the chi-square distribution to find the probability of observing a sample standard deviation as extreme as or more extreme than the one obtained, assuming the null hypothesis is true.
In this case, the P-value is 0.343. This means that if the null hypothesis is true, there is a 34.3% chance of obtaining a sample standard deviation of 5.985 or more extreme. Since the P-value is greater than the common significance level of 0.05, we fail to reject the null hypothesis. Therefore, we do not have strong evidence to suggest that the population standard deviation is less than 7.2.
In conclusion, the correct choice is: The P-value 0.343 is not significant and does not strongly suggest that sigma is less than 7.2.
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A carpenter makes bookcases in 2 sizes, large and small. It takes 4 hours to make a
large bookcase and 2 hours to make a small one. The profit on a large bookcase is
$35 and on a small bookcase is $20. The carpenter can spend only 32 hours per
week making bookcases and must make at least 2 of the large and at least 4 of the
small each week. How many small and large bookcases should the carpenter make
to maximize his profit? What is his profit?
Answer:
6 large and 4 small
Step-by-step explanation:
6 times 4 =242 time 4= 832 hoursplease help with this?!?
Answer:
196.1
Step-by-step explanation:
Area of a circle is [tex]\pi r^{2}[/tex] so in order to find the radius you divide the diameter by 2 to get 7.9
Then you do [tex]7.9^{2}[/tex] x [tex]\pi[/tex] to get around 196.1
PLEASE SOMEONE HELPPPPPPP
Answer:
12, 25, 26, 26, 26, 34, 35, 39, 42, 42, 50, 72.
Step-by-step explanation:
A stem and leaf plot works like a digit separator. The left is the first number, which is usually repeated, and the right is the number you add to it.
In this example, 3 is used three times for the numbers 34, 35, and 39.
PLEASEEEEEEEEEE HELPPPPPPPPPPPPP
Answer:
i dont kno bestie.. :/
Step-by-step explanation:
An agronomist measures the lengths of n = 26 ears of corn. The mean length was 31.5 cm and the standard deviation was s= 5.8 cm. Find the Upper Boundary for a 95% confidence interval for mean length of corn ears. O 57.5 29.2 O 0.05 O 33.8
The upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm
To find the upper boundary for a 95% confidence interval for the mean length of corn ears, we can use the formula:
Upper Boundary = Mean + (Critical Value * Standard Error)
The critical value corresponds to the desired level of confidence. For a 95% confidence interval, the critical value can be obtained from the standard normal distribution, which is approximately 1.96.
The standard error is calculated by dividing the standard deviation by the square root of the sample size:
Standard Error = s / [tex]\sqrt{(n)}[/tex]
Given that the mean length was 31.5 cm (Mean) and the standard deviation was s = 5.8 cm, and the sample size was n = 26, we can calculate the upper boundary as follows:
Standard Error = 5.8 / [tex]\sqrt{26}[/tex] ≈ 1.138
Upper Boundary = 31.5 + (1.96 * 1.138) ≈ 33.8
Therefore, the upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm.
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If y varies directly as x, and y = 6 when x = 4, find y when x = 12.
y =
y=14 I hope this helps!!
Kim is repainting a storage trunk shaped like a rectangular prism as shown.
Kim will paint all the faces of the outside of the storage trunk when it is closed. How many square feet will Kim paint?
Answer:
i got 54ft^2
Step-by-step explanation:
1. Prove that, for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1
The statement " for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1" is proved.
If η is the Euler totient function defined by η(n)=n * (1-1/p1) * (1-1/p2) * ....* (1-1/pk) then for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1.
To prove η 2 n(n+1) Σκ Σ 2 k=1 for every integer n > 1 we have to solve the given question :
1) We know that η(n) = n * (1-1/p1) * (1-1/p2) * ....* (1-1/pk).and
let S = Σκ Σ 2 k=1
2) For n = 2 we have η(2) = 2 * (1 - 1/2) = 1
Hence, S = Σκ Σ 2 k=1 = 1*2=2
Now, η(4) = 4 * (1 - 1/2)(1 - 1/2) = 2 and η(6) = 6 * (1 - 1/2)(1 - 1/3) = 2
Therefore, η 2 n(n+1) Σκ Σ 2 k=1
Hence, S = Σκ Σ 2 k=1 = 2* (2 + 1) * 2 = 12.
3) For n=3, we haveη(3) = 3 * (1 - 1/3) = 2S = Σκ Σ 2 k=1 = 1 * 2 + 2 * 3 = 8
Also, η(6) = 6 * (1-1/2)(1-1/3) = 2
Hence, η 2 n(n+1) Σκ Σ 2 k=1
Thus, S = Σκ Σ 2 k=1 = 2* (3 + 1) * 2 = 16
Therefore, for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1.
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A restaurant sells an 8-oz drink for $2.56 and a 12 oz drink for $3.66. Which drink is the better buy? i need help fast :(
Answer:
12 oz
Step-by-step explanation:
2.56 ÷ 8 = 0.32 per oz
3.66 ÷ 12= 0.305 per oz
Please Help. What expression is equivalent to 6( t - 5 ) + 3
A. 6t - 2
B. 6t - 12
C. 3 ( 2t - 11 )
D. 3 ( 2t + 9 )
Use the normal distribution of SAT critical reading scores for which the mean is 505 and the standard deviation is 118. Assume the variable x is normally distributed. (a) What percent of the SAT verbal scores are less than 600? (b) If 1000 SAT verbal scores are randomly selected, about how many would you expect to be greater than 575? Click to view page 1 of the standard normal table. Click to view page 2 of the standard normal table. (a) Approximately 79 % of the SAT verbal scores are less than 600. (Round to two decimal places as needed.) (b) You would expect that approximately 722 SAT verbal scores would be greater than 575.
Therefore, we would expect that approximately 722 SAT verbal scores out of 1000 would be greater than 575.
For a normal distribution of SAT critical reading scores with a mean of 505 and a standard deviation of 118, approximately 79% of the SAT verbal scores are less than 600. If 1000 SAT verbal scores are randomly selected, it is expected that approximately 722 of them would be greater than 575.
To determine the percentage of SAT verbal scores that are less than 600, we need to find the area under the normal distribution curve to the left of 600. We can use the standard normal distribution table or a statistical software to find the corresponding z-score.
First, we calculate the z-score using the formula:
z = (x - μ) / σ
Substituting the values:
z = (600 - 505) / 118
z ≈ 0.8051
Using the standard normal distribution table, we can find the area to the left of z = 0.8051, which is approximately 0.7910.
To determine the percentage, we multiply the result by 100, giving us approximately 79% of SAT verbal scores that are less than 600.
For part (b), we can apply the same approach. We calculate the z-score for x = 575:
z = (575 - 505) / 118
z ≈ 0.5932
Using the standard normal distribution table, we find the area to the left of z = 0.5932, which is approximately 0.7242. This means that approximately 72.42% of SAT verbal scores are less than 575.
To estimate the number of SAT verbal scores greater than 575 in a sample of 1000, we multiply the percentage by the sample size:
Number of scores greater than 575 = 0.7242 * 1000 ≈ 722.
Therefore, we would expect that approximately 722 SAT verbal scores out of 1000 would be greater than 575.
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Pls help and if you can show me how you do it :)
Find the number less than 40, that is
divisible by 5, and when divided by 6
has a remainder of 2.
A line is graphed on the coordinate plane below. Another line y = -x + 2
will be graphed on the same coordinate plane to create a system of equations.
What is the solution to that system of equations?
A. (-2,4)
B. (0,-4)
C. (2,-4)
D. (4,-2)
The solution to the given system of equations y = -x + 2 are option A. (-2,4) and D. (4,-2).
What is a system of equations?A system of equations is two or more equations that can be solved together to get a unique solution. the power of the equation must be in one degree.
The equation is given as
y = -x + 2
here, we need to find the solutions to the equation, we can apply the given options one by one to satisfy the equation.
For the solution
A. (-2,4)
y = -x + 2
Substitute the value x = -2 and y = 4
y = 4
-x + 2 = -(-2) + 2 = 4
Thus, the given solution are the system of equation.
For the solution
B. (0,-4)
y = -x + 2
Substitute the value x = 0 and y = -4
y = -4
-x + 2 = 0 + 2 = 2
Thus, both the sides are not equal so, the given solution are not the system of equation.
For the solution
C. (2,-4)
y = -x + 2
Substitute the value x = 2 and y = -4
y = -4
-x + 2 = 2 + 2 = 4
Thus, both the sides are not equal so, the given solution are not the system of equation.
For the solution
D. (4,-2)
y = -x + 2
Substitute the value x = 4 and y = -2
y = -2
-x + 2 = -4 + 2 = -2
Thus, the given solution are the system of equation.
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Consider the region in the xy-plane bounded from above by the curve y=4x−x^2 and below by the curve y=x. Find the centroid of the region. (i.e. the center of mass of this region if the mass density is p =1)
The centroid of the region bounded from above by the curve y = 4x - x² and below by the curve y = x is (2/3, 4/3).
The region is bounded from above by the curve y = 4x - x² and below by the curve y = x. We need to find the points of intersection between these two curves. Setting the equations equal to each other,
4x - x² = x
Rearranging,
x² - 3x = 0
Factoring,
x(x - 3) = 0
So, x = 0 or x = 3.
The region is bounded from x = 0 to x = 3. To find the y-values within this region, we evaluate the equations y = 4x - x² and y = x at these x-values.
For x = 0,
y = 4(0) - (0)² = 0
For x = 3,
y = 4(3) - (3)² = 12 - 9 = 3
Thus, the y-values within the region are y = 0 to y = 3. Now, we calculate the area of the region by integrating the difference of the upper and lower curves,
A = ∫[0,3] [(4x - x²) - x] dx
A = ∫[0,3] (3x - x²) dx
A = [3x²/2 - x³/3] evaluated from x = 0 to x = 3
A = [27/2 - 9/3] - [0 - 0]
A = [27/2 - 3] - 0
A = 21/2
Now, for the centroid,
x = (1/A) * ∫[0,3] x * [(4x - x²) - x] dx
Simplifying,
x = (1/A) * ∫[0,3] (3x² - x³) dx
x = (1/A) * [x³ - x⁴/4] evaluated from x = 0 to x = 3
x = (1/A) * [(3)³ - (3)⁴/4] - [0 - 0]
x = (1/A) * [(27) - (81)/4] - 0
x = (1/A) * [(108 - 81)/4]
x = (1/A) * (27/4)
x = 27/(4A)
x = 27/(4 * 21/2)
x = 2/3, and,
x = (1/A) * ∫[0,3] [(4x - x²) - x]² dx
Simplifying,
y = (1/A) * ∫[0,3] (16x² - 8x³ + x⁴) dx
y = (1/A) * [(16x³/3 - 8x⁴/4 + x⁵/5)] evaluated from x = 0 to x = 3
y = (1/A) * [(16(3)³/3 - 8(3)⁴/4 + (3)⁵/5)] - [0 - 0]
y = (1/A) * [(16 * 27/3 - 8 * 81/4 + 243/5)]
y = (1/A) * [(144/3 - 648/4 + 243/5)]
y = (1/A) * [(480 - 972 + 243)/60]
y = (1/A) * (480 - 972 + 243)/60
y = -83/(20A)
Since A = 21/2, we can substitute it in,
y = -83/(20 * 21/2)
y = -83/(210/2)
y = -83/(105)
y = -4/5
Therefore, the centroid of the region is (2/3, 4/3).
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Use the method of variation of parameters to find a particular solution of the following differential equation. y'' - 12y' + 36y = 10 e 6x What is the Wronskian of the independent solutions to the homogeneous equation? W(71.72) = The particular solution is yp(x) =
The Wronskian of the autonomous answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.The specific arrangement is yp(x) = 5x e^(6x) (2 - x)The Wronskian of the free answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.
The differential equation is y'' - 12y' + 36y = 10 e 6x. We need to use the method of parameter variation to find the particular solution to the given differential equation. Let's begin by resolving the homogeneous differential equation. The homogenous piece of the differential condition isy'' - 12y' + 36y = 0The trademark condition is r² - 12r + 36 = 0 which can be figured as (r - 6)² = 0So, the arrangement of the homogenous piece of the differential condition is given byy_h(x) = c1 e^(6x) + c2 x e^(6x)where c1 and c2 are inconsistent constants. Presently, let us find the specific arrangement of the given differential condition utilizing the strategy for variety of boundaries. Specific arrangement of the given differential condition isy_p(x) = - y1(x) ∫(y2(x) f(x)/W(x)) dx + y2(x) ∫(y1(x) f(x)/W(x)) dxwhere, y1 and y2 are the arrangements of the homogeneous condition, W is the Wronskian of the homogeneous condition and f(x) is the non-homogeneous term of the differential condition. Hence, y_p(x) = -e(6x) (x e(6x) / e(12x)) dx + x e(6x) (e(6x) (10 e(6x)) / e(12x)) dx = -e(6x) (10x) dx + x e(6x) (10) dx = -5 That's what we know, W(x) = | y1 y2 | | y1' y2' | = e^(12x)Therefore, W(71.72) = e^(12*71.72) = 6.06 × 10²⁸Hence, the Wronskian of the autonomous answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.The specific arrangement is yp(x) = 5x e^(6x) (2 - x)The Wronskian of the free answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.
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In the exercise, X is a binomial variable with n = 6 and p = 0.4. Compute the given probability. Check your answer using technology. HINT [See Example 2.] (Round your answer to five decimal places.)
P(X ≤ 2)=?
To compute the probability P(X ≤ 2) for a binomial variable X with n = 6 and p = 0.4, we need to sum the probabilities of X taking on the values 0, 1, and 2.
[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
To calculate these probabilities, we can use the binomial probability formula:
P(X = k) = (n choose k) [tex]* p^k * (1 - p)^(n - k)[/tex]
where (n choose k) represents the binomial coefficient, given by (n choose k) = n! / (k! * (n - k)!)
Let's calculate the probabilities step by step:
P(X = 0) = (6 choose 0) * [tex]0.4^0 * (1 - 0.4)^(6 - 0)[/tex]
P(X = 1) = (6 choose 1) * [tex]0.4^1 * (1 - 0.4)^(6 - 1)[/tex]
P(X = 2) = (6 choose 2) * [tex]0.4^2 * (1 - 0.4)^(6 - 2)[/tex]
Using the binomial coefficient formula, we can calculate the probabilities:
P(X = 0) = 1 * 1 * [tex]0.6^6[/tex] ≈ 0.04666
P(X = 1) = 6 * 0.4 * [tex]0.6^5[/tex] ≈ 0.18662
P(X = 2) = 15 * [tex]0.4^2 * 0.6^4[/tex] ≈ 0.31104
Now, let's sum these probabilities to find P(X ≤ 2):
P(X ≤ 2) ≈ 0.04666 + 0.18662 + 0.31104 ≈ 0.54432
Therefore, the probability P(X ≤ 2) is approximately 0.54432.
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simplify leaving your answer in the standard form
[tex] \frac{0.0225 \times 0.0256}{0.0015 \times 0.48} [/tex]
Answer:
0.8 is the standard form
5 in = ___________ ft *Write your answers like this: whole number, one space, numerator, /, denominator. Example: 1 1/2 * PLEASE AWNSER FAST <3
Answer:
0.416667 ft
Step-by-step explanation:
What is the answer to this question?
what is the approximate radius of a sphere with a volume of 900 cm squared
A 12 cm
B 36 cm
C 18cm
D 6cm
Answer:
about 5.99 or D. 6 cm
Step-by-step explanation:
you can use this formula
[tex]V=4/3 * \pi *r^{3}[/tex]
what is 1/3 plus 1/2 in fraction form
Answer:
5/6
Step-by-step explanation:
Hope this helped!!!
Need help with all the question
Answer:
Step-by-step explanation:
So in ratios you can mostly all of the time scale your answer. So by determining how much increase there is in the baby's thigh bone each week you can pretty much answer these questions.
keep in mind: Proportional means having the same ratio. A scale factor is the ratio of the model measurement to the actual measurement in simplest form.
Example from https://www.mathsisfun.com/numbers/ratio.html
A ratio says how much of one thing there is compared to another thing.
ratio 3:1
There are 3 blue squares to 1 yellow square
Ratios can be shown in different ways:
Use the ":" to separate the values: 3 : 1
Or we can use the word "to": 3 to 1
Or write it like a fraction: 31
A ratio can be scaled up:
ratio 3:1 is also 6:2
Here the ratio is also 3 blue squares to 1 yellow square,
even though there are more squares.
find the volume of the solid that results when the region bounded by y=x−−√, y=0 and x=36 is revolved about the line x=36.
The volume of the solid obtained by revolving the region bounded by y = x - √x, y = 0, and x = 36 around the line x = 36 can be found using the method of cylindrical shells. The resulting volume is approximately 3,012 cubic units.
To calculate the volume, we integrate the formula for the volume of a cylindrical shell, which is given by V = 2π∫[a,b] x * h(x) dx, where [a,b] represents the range of x values.
In this case, the lower bound of integration is 0 and the upper bound is 36, since the region is bounded by y = 0 and x = 36. The height of the cylindrical shell, h(x), is given by the difference between the x-coordinate of the curve y = x - √x and the line x = 36.
To obtain the x-coordinate of the curve, we set x - √x = 0 and solve for x. This gives us x = 0 or x = 1.
Next, we calculate the difference between x and 36, which gives us the height of the cylindrical shell. Then, we substitute the expressions for x and h(x) into the volume formula and integrate with respect to x.
After performing the integration, we find that the volume of the solid is approximately 3,012 cubic units.
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it always tells me i have to put 20 characters but i really need help
Answer:
B
Step-by-step explanation:
25/100 is 25%.
. **y" + xy' + y = 0, y(t) = 3 . y'(1)=4 (12pts) 3. Solve the Cauchy-Euler IVP:
The solution to the Cauchy-Euler initial value problem is -3/2
To solve the Cauchy-Euler initial value problem, we need to find the general solution of the differential equation and then use the initial conditions to determine the specific solution.
The given Cauchy-Euler differential equation is:
y" + xy' + y = 0
To solve this equation, we assume a solution of the form [tex]y(x) = x^r[/tex]
Differentiating twice with respect to x, we have:
[tex]y' = rx^{r-1}[/tex] and y" = [tex]r(r-1)x^{r-2}[/tex]
Substituting these expressions into the differential equation, we get:
[tex]r(r-1)x^{r-2} + x(rx^{r-1}) + x^r = 0[/tex]
[tex]r(r-1)x^{r-2} + r*x^r + x^r = 0[/tex]
[tex]x^{r-2}(r(r-1) + r + 1) = 0[/tex]
For a non-trivial solution, the expression in parentheses must equal zero:
r(r-1) + r + 1 = 0
Expanding and rearranging, we have:
[tex]r^2 - r + r + 1 = 0\\r^2 + 1 = 0[/tex]
The roots of this equation are complex numbers:
r = ±i
Therefore, the general solution of the Cauchy-Euler differential equation is:
[tex]y(x) = c_1x^i + c_2x^{-i}[/tex]
To simplify the solution, we can rewrite it using Euler's formula:
[tex]y(x) = c_1x^i + c_2x^{-i}\\ = c_1(cos(ln(x)) + i*sin(ln(x))) + c_2(cos(ln(x)) - i*sin(ln(x)))\\ = (c_1 + c_2)cos(ln(x)) + (c_1 - c_2)i*sin(ln(x))[/tex]
Now, let's apply the initial conditions to find the specific solution. We are given:
y(t) = 3 and y'(1) = 4
Substituting x = t into the solution, we have:
[tex](c_1 + c_2)cos(ln(t)) + (c_1 - c_2)i*sin(ln(t)) = 3[/tex]
To satisfy this equation, the real parts and imaginary parts on both sides must be equal.
From the real parts:
[tex](c_1 + c_2)cos(ln(t)) = 3[/tex]
From the imaginary parts:
[tex](c_1 - c_2)i*sin(ln(t)) = 0[/tex]
Since sin(ln(t)) ≠ 0 for any t, we must have ([tex]c_1 - c_2[/tex]) = 0.
This implies [tex]c_1 = c_2[/tex].
Substituting [tex]c_1 = c_2[/tex] into the real part equation, we get:
[tex]2c_1cos(ln(t)) = 3[/tex]
Solving for [tex]c_1[/tex], we find:
[tex]c_1 = 3/(2cos(ln(t)))[/tex]
Therefore, the specific solution of the Cauchy-Euler initial value problem is:
y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))
Now, we can find y'(1) by differentiating the specific solution with respect to x and evaluating it at x = 1:
y'(x) = -(3/2)(ln(t)sin(ln(x)) + cos(ln(x)))
y'(1) = -(3/2)(ln(t)sin(ln(1)) + cos(ln(1)))
= -(3/2)(ln(t)(0) + 1)
= -3/2
Therefore, the solution to the Cauchy-Euler initial value problem is:
y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))
y(t) = 3
y'(1) = -3/2
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