Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.
Answers
The basis of solutions using the Frobenius method for the differential equation xy" + (1 – 2x)y' + (x - 1)y = 0 is y₁(x) = a₀x⁻ⁿ, y₂(x) = aₙx⁻ⁿ, and y₃(x) = aₙ₊₁x²⁻ⁿ, where a₀, aₙ, and aₙ₊₁ are constants and n is a non-negative integer.
To find a basis of solutions using the Frobenius method, we assume that the solution can be expressed as a power series:
y(x) = ∑[n=0 to ∞] aₙ\(x^{n+r}\),
where aₙ represents the coefficients and r is a constant to be determined. Let's begin by differentiating the series:
y'(x) = ∑[n=0 to ∞] aₙ(n+r)\(x^{n+r-1}\),
y''(x) = ∑[n=0 to ∞] aₙ(n+r)(n+r-1)\(x^{n+r-2}\).
Substituting these derivatives into the given differential equation, we have:
xy" + (1 – 2x)y' + (x - 1)y = 0
∑[n=0 to ∞] aₙ(n+r)(n+r-1)\(x^{n+r}\) + ∑[n=0 to ∞] aₙ(n+r)\(x^{n+r}\) - 2x∑[n=0 to ∞] aₙ(n+r)\(x^{n+r-1}\) + ∑[n=0 to ∞] aₙ\(x^{n+r}\) - ∑[n=0 to ∞] aₙ\(x^{n+r}\) = 0.
∑[n=1 to ∞] aₙ(n+r-1)\(x^{n+r}\) = ∑[n=0 to ∞] aₙ(n+r)\(x^{n+r}\) * \(x^r\).
Finally, for the third term, we'll differentiate the series:
-2x∑[n=0 to ∞] aₙ(n+r)\(x^{n+r-1}\) = -2x∑[n=0 to ∞] aₙ(n+r)\(x^n * x^r\).
Now, we can combine the terms:
∑[n=0 to ∞] (aₙ(n+r)(n+r-1) + aₙ(n+r) - 2aₙ\((n+r))x^n * x^r\) = 0.
Since this equation should hold for all values of x, the coefficients of each power of x must be zero. Therefore, we obtain the following recurrence relation:
aₙ(n+r)(n+r-1) + aₙ(n+r) - 2aₙ(n+r) = 0.
Simplifying the equation, we have:
aₙ[(n+r)(n+r-1) + (n+r) - 2(n+r)] = 0.
aₙ(n+r)(n+r-1+1-2) = 0.
aₙ(n+r)(n+r-2) = 0.
Now, we have three cases to consider:
Case 1: aₙ = 0
If aₙ = 0, then the coefficient of \(x^n\) is zero, and we can move on to the next value of n.
Case 2: n+r = 0
If n+r = 0, then r = -n. In this case, the solution becomes:
y₁(x) = a₀\(x^r\) = a₀\(x^{-n}\).
Case 3: (n+r)(n+r-2) = 0
If (n+r)(n+r-2) = 0, then we have two solutions:
Solution 1: n+r = 0
In this case, r = -n. The solution becomes:
y₂(x) = aₙ\(x^r\) = aₙ\(x^{-n}\).
Solution 2: n+r-2 = 0
In this case, r = 2-n. The solution becomes:
y₃(x) = aₙ₊₁\(x^r\) = aₙ₊₁\(x^{2-n}\).
Therefore, we have three linearly independent solutions:
y₁(x) = a₀\(x^{-n}\),
y₂(x) = aₙ\(x^{-n}\),
y₃(x) = aₙ₊₁\(x^{2-n}\).
We can identify the series expansions of known functions as follows:
y₁(x) = a₀\(x^{-n}\) is a series expansion of a constant function.
y₂(x) = aₙ\(x^{-n}\) is a series expansion of \(x^{-n}\), which is a generalized power function.
y₃(x) = aₙ₊₁\(x^{2-n}\) is a series expansion of \(x^{2-n}\), which is another generalized power function.
These solutions form a basis for the given differential equation using the Frobenius method.
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Related Questions
point d is the interior of angleABE and point E is the interior of angleDBC....(press on the picture please) (also i scratched out where i tried to work it out)
Answers
Answer:
c)41
Step-by-step explanation:
∠DBE = (∠ABE +∠DBC) -∠ABC
= (74 + 103) -136
= 177 - 136
= 41
BRAINLIEST ASAP AND 15 POINTS Sawyer is creating a scaled drawing of the pentagon for his social studies class. The Pentagon is a regular polygon with each side measuring 921 feet long. He plans to use a scale of 3 cm = 307 feet. What will be the total perimeter of the pentagon on sawyer's drawing?
Answers
Answer:
\(9\text{ cm}\)
Step-by-step explanation:
We can write the following proportion:
\(\frac{\text{old perimeter}}{\text{scaled perimeter}}=\frac{\text{old scale factor}}{\text{new scale factor}}\)
We know that the original perimeter of the Pentagon is 921 feet. So, substitute 921 for "old perimeter."
We are going to use a scale of 3 cm = 307 feet. Since the 307 feet corresponds to the original perimeter, we are going to substitute 307 for "old scale factor" and 3 for "new scale factor."
This gives us:
\(\frac{921}{x}=\frac{307}{3}\)
So, to find our scaled perimeter, we just need to find x.
Cross multiply:
\(307x=2763\)
Divide both sides by 307:
\(x=9\)
In other words, the total perimeter of Sawyer's drawing will be 9cm.
And we're done!
-a to the third power - b to the second power if a= -3 and b= -4
Answers
Answer:
-a Would be -27 and -b would be -16
Step-by-step explanation:
if one full-time student who is a freshman or sophomore is selected at random, what is the probability that the student will be a student who lives on campus?
Answers
The probability that the student will be a student who lives on campus is 0.24
In math the term probability is referred as the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions.
Here we have given that one full-time student who is a freshman or sophomore is selected at random.
Here we need to find the probability that the student will be a student who lives on campus.
Here we know that the probability that a student is a sophomore is calculated as,
=> P(S) = 19/42 = 0.45.
Whereas the probability that a student has a freshman is calculated as,
=>P(H)=25/42=0.6.
Then the probability that the student will be a student who lives on campus is calculated as,
=> P(H∩S)=P(S)+P(H)−P(S∪H)
Apply the values then simplify this one then we get,
=> 0.45+0.6−(42−8)/42=0.24
Complete Question:
Suppose that a certain college class contains 42 students and then if one full-time student who is a freshman or sophomore is selected at random, what is the probability that the student will be a student who lives on campus?
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Given: Tangent (StartFraction x Over 2 EndFraction) = StartFraction 1 minus cosine (x) Over sine (x) EndFraction
Prove: Tangent (StartFraction x Over 2 EndFraction) = StartFraction sine (x) Over 1 + cosine (x) EndFraction
Complete the steps of the proof.
♦:
♣:
answer is multiply by form of 1, pythagorean identity
Answers
We can write tan(x/2) as -
tan(x/2) = [1 - cos(x)]/sin(x) = sin(x)/(1 + cos(x))
What are trigonometric functions?In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. Example -
sine, cosine, tangent etc
Given is -
tan(x/2) = [1 - cos(x)]/sin(x)
We have -
tan(x/2) = [1 - cos(x)]/sin(x)
tan(x/2) = (1 - cos(x))(1 + cos x) / sin(x)(1 + cos(x))
tan(x/2) = (1 - cos²x) / sin(x)(1 + cos(x))
tan(x/2) = sin²(x) / sin(x)(1 + cos(x))
tan(x/2) = sin(x)/(1 + cos(x))
Hence proved.
Hence, we can write tan(x/2) as -
tan(x/2) = [1 - cos(x)]/sin(x) = sin(x)/(1 + cos(x))
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Answer:
multiply by a form of 1, pythagorean identity
Step-by-step explanation:
on edge
Help pls. For math. 15 points
Answers
Answer:
y=mx+b
so
y=8x+3. the second answer choice
Answer:
y=8x+3
Step-by-step explanation:
we are given the slope (m) as 8 and the y intercept (b) as 3.
The given options are in y=mx+b format.
Use your given information to write the equation in y=mx+b form
we know that m (slope) is 8. The equation so far is y=8x+b
we know that b (y intercept) is 3. So the equation of the line is y=8x+3
Hope this helps!
Find the eigenvalues and eigenvectors of A geometrically over the real numbers R. (If an eigenvalue does not exist, enter DNE. If an eigenvector does not exist, enter DNE in any single blank.) A = 1 0 (reflection in the line y x) 0 1 = -1 has eigenspace span (smaller A-value) = 1 has eigenspace span (largerA-value)
Answers
The matrix A represents the linear transformation that reflects points across the line y=x in the Cartesian plane.
To find the eigenvalues of A, we solve the characteristic equation:
det(A - λI) = 0
where I is the identity matrix and λ is an eigenvalue of A.
A - λI =
[1-λ 0]
[0 1-λ]
det(A - λI) = (1-λ)(1-λ) - 0*0 = (1-λ)^2
Setting this expression equal to zero and solving for λ, we find:
(1-λ)^2 = 0
1-λ = 0
λ = 1
So the only eigenvalue of A is 1.
To find the eigenvectors corresponding to λ=1, we solve the system of equations:
(A - λI)v = 0
where v is an eigenvector of A.
For λ=1, we have:
(A - I)v =
[0 0]
[0 0]
which implies that any non-zero vector in the plane (i.e., any non-zero vector in R^2) is an eigenvector of A corresponding to the eigenvalue 1.
Therefore, the eigenspace corresponding to the eigenvalue 1 is all of R^2, and any non-zero vector in R^2 can be taken as an eigenvector.
In summary, the eigenvalue of A is 1, and the eigenspace corresponding to this eigenvalue is all of R^2. Any non-zero vector in R^2 can be taken as an eigenvector.
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A is the center of one circle, and B is the
center of the other. Explain how we know triangle ABC
is equilateral.
Answers
Answer:
a
Step-by-step explanation:
because it goes to one side to another
8x-2=46 what is this but like you have to use a upside down T
Answers
Step-by-step explanation:
8x-2=46
collect like terms
8x=46+2
8x=48
Divide both sides by 8
8x÷8=48÷8
x=6
x=6
Step by step:
Help me please!!!!!!!
Answers
Answer:
you need to find out how many miles per hour irst then see how many itll be in 24 hours
Step-by-step explanation:
Answer:
D = 6.2t, where t is in hours
Now, solve for 24 hours for t.
D = 6.2(24) = 148.8 miles in 24 hours.
Step-by-step explanation:
First, find out how far the butterfly can fly in 1 hour.
93 miles / 15 hours = 6.2 miles per hour
Let D = the distance in miles the butterfly can fly in t hours.
Your equation will be
D = 6.2t, where t is in hours.
The ratio of Cheko's books to Miko's books is 5 : 12. Miko has 48 books. How many books must Miko give to Cheko so that both will have the same number of books?
Answers
Answer:
Step-by-step explanation:
Cheko's books (C) to Miko's books (M) is 5 : 12
C/M = 5/12
C = (5/12)M
We know M=48, so:
C = (5/12)(48)
C = 5*(48/12)
C = 5*4
C = 20 [Cheko has 20 books]
We want M=C, so Miko must give Cheko x number of books so that C/M is (1/1) or 1.
C/M is currently 20/48
We want C/M = (20+x)/(48-x) = 1
(20+x)/(48-x) = 1
(20+x) = (48-x)
2x = 28
x = 14
Once Miko gives Cheko 14 books, they each have the same number: (48-14)/(20+14) = 34/34. Everyone's happy. [We don't know WHICH books Miko chose to give to Cheko. Rumor is all 14 were math books]
4. the sides of a triangle are in the extended ratio 2:6:7. if the perimeter of the triangle is 45 inches, then what is the length of the shortest side?
5. the measures of the angles of a triangle are in the extended ratio 17:16:12. what is the measure of the largest angle?
6. what is the value of X in the proportion? \(\frac{2x+7}{3x-4}\)=\(\frac{11}{2}\)
Answers
Step-by-step explanation:
4. 2/ 15 x 45/1 = 90/15
=6inches
( 15 is the total of the ratio)
5. Sum of angles in a triangle in 180
Total of the ratio is 17+16+12 = 45
Largest angle is 17/ 45 x180/1 = 68 degree
Ans 68degrees
6. 2x + 7/ 3x - 4 = 11/2
Cross multiply
2( 2x + 7) = 11(3x - 4)
Expand the equation
4x +14 = 33x - 44
Subract 4x from both sides
14 = 29x - 44
Add 44 to both sides
58 =29x
Divide both sides by 29
X =2
Mark's science club sold brownies and cookies to raise money for a trip to the natural history museum.
Answers
The prices for each item are given as follows:
Brownies: $1.25.Cookies: $0.75.How to obtain the prices?The prices are obtained with a system of equations, for which the variables are given as follows:
Variable x: cost of a brownie.Variable y: cost of a cookie.From the first row of the table, we have that:
40x + 32y = 74.
Simplifying by 32, we have that:
1.25x + y = 2.3125
y = 2.3125 - 1.25x.
From the second row, we have that:
20x + 25y = 43.75.
Replacing the first equation into the second, the value of x is obtained as follows:
20x + 25(2.3125 - 1.25x) = 43.75
11.25x = 14.0625
x = 14.0625/11.25
x = 1.25.
Then the value of y is obtained as follows:
y = 2.3125 - 1.25(1.25)
y = 0.75.
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Please help asap I put a photo
For each relation, decide whether or not it is a function.
Answers
Answer:
1: Function
2: Function
3: Not a function
4: Function
Step-by-step explanation:
1: Not a function
2: Function
3: Not a function
4: Function
REMINDER: If a x value has two diffreent y values, it is NOT a function!!!!
Have an amazing day!!
PLEASE RATE AND MARK BRAINLIEST!!!!
Please help me answer this
Answers
Answer:
Step-by-step explanation:
It's B. If you divide 15 and 12 ( The first triangle) it's 1.25. So, for the second triangle, do 5 x 1.25 and you get 6.25
the standard error of estimate in a simple linear regression is equal to: a) the square root of the mean squared error. b) the regression sum of squares divided by the total sum of squares.
Answers
The standard error of estimate in a simple linear regression is equal to: a) the square root of the mean squared error.
What is standard Error of Estimate?Standard error of estimate simply refers to the amount of mistake that can be seen when the estimate from the regression analysis is utilized instead of the actual data, and a small estimate error is not necessarily a bad thing.
A. The assumption about errors at different levels is that the variance between two errors is the same as the variance between two other errors. The variance basically refers to the residual. The standard error is computed assuming homoskedasticity, that is, the constant variance of the error terms.
B. The standard error of the estimate is a way to measure the accuracy of the predictions made by a regression model.
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g(n)=n² +4n
h(n)=-3n-3
Find (g+h)(n-4)
Answers
The function (g+h)(n-4) is the sum of the function g(n-4) and h(n-4) will be written as (g+h)(n-4) = n² - 7n + 9.
What is a function?A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.
The functions are given below.
g(n) = n² + 4n
h(n) = - 3n - 3
The function (g+h)(n) is calculated as,
(g+h)(n) = g(n) + h(n)
(g+h)(n) = n² + 4n - 3n - 3
(g+h)(n) = n² + n - 3
Put n = n - 4, then we have
(g+h)(n-4) = (n - 4)² + (n - 4) - 3
(g+h)(n-4) = n² + 16 - 8n + n - 4 - 3
(g+h)(n-4) = n² - 7n + 9
The function (g+h)(n-4) will be (g+h)(n-4) = n² - 7n + 9.
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what is the given domain for this
y=-2+2
Answers
Answer:
Y = 0
Step-by-step explanation:
Y = -2 + 2
Negative 2 plus 2 equals zero, there is no more numbers so Y = 0.
Please help me on 6,7and8
Answers
Consider the given probability histogram of a binomial random variable. A histogram titled binomial random variable (n = 5, p = 0. 5) has x on the x-axis and probability on the y-axis. 0, 0. 03; 1, 0. 15; 2, 0. 3; 3, 0. 3; 4, 0. 15; 5, 0. 3. What are the center and shape of the distribution? Center: 2 Shape: symmetric Center: 2. 5 Shape: symmetric Center: 2. 5 Shape: slightly skewed Center: 3 Shape: uniform.
Answers
The shape of the distribution is symmetric. In summary, the correct answer is: Center: 2.5, Shape: symmetric.
To determine the center and shape of the distribution represented by the given probability histogram of a binomial random variable, we need to analyze the data.
The center of the distribution refers to the average or mean value of the random variable. In this case, we can calculate the mean by multiplying each value of x by its corresponding probability and summing them up.
Mean = (0 * 0.03) + (1 * 0.15) + (2 * 0.3) + (3 * 0.3) + (4 * 0.15) + (5 * 0.3) = 2.5
So, the center or mean of the distribution is 2.5.
To determine the shape of the distribution, we can observe the relative heights of the bars in the histogram. In a symmetric distribution, the probabilities of the values on one side of the center are the same as the probabilities on the other side. In this case, the probabilities are symmetrical around the center value of 2.5.
Therefore, the shape of the distribution is symmetric.
In summary, the correct answer is:
Center: 2.5
Shape: symmetric
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The sum of 5 and the twice a number is at most 27
Answers
Answer:
Step-by-step explanation:
let the number=x
5+2x≤27
2x≤25-5
2x≤20
x≤4
can someone pls help with the answers i don’t need explanation just answers pls
Answers
Answer:
4) 74
5) 114
6) 124
7) 96
8) 64
Step-by-step explanation:
Please help me out :)
Answers
NEED HELP ASAP
If the side lengths of a cube are 15 feet, what is the correct way to write the expression to represent the volume of the cube in exponential form?
15 ⋅ 15 ⋅ 15
15 ⋅ 3
3^15
15^3
Answers
2. 45
3. 14348907
4. 3375
but the 3 or 2 ???
It costs twelve dollars to get in to the fair. Tickets for rides cost extra and are d dollars each. If Tyra buys 20 tickets, how much would it cost her for the fair?
Answers
Answer:
12 + 20d
Step-by-step explanation:
OG fare = 12
Ticket price =d
20 Tickets =20d
Answer detailed please
Answers
Answer:
container A
Step-by-step explanation:
In Container A, we can plot the points (0,60) adn (2, 35)
The slope is :
\(m_A = \frac{y_2-y_1}{x_2-x_1} \\\\= \frac{35-60}{2-0} \\\\= \frac{-25}{2}\\ \\m_A = -12.5\\\)
For container B, the slope is:
\(m_B = \frac{y_2-y_1}{x_2-x_1} \\\\= \frac{32-54}{3-1} \\\\= \frac{-22}{2}\\ \\m_B = -11\\\)
The negative sign of the slope indicates the direction of the slope
\(|m_A| = 12.5\\\\|m_B| = 11\)
12.5 > 11
The slope of container A is steeper than container B
Therefore, the water is draining out of container A at a faster rate than container B
Solve the System of Equations
-5x+4y=3
x=2y-15
Answers
Answer:
Point Form:
(9,12)
Equation Form:
x = 9
y = 12
Step-by-step explanation:
Answer:
\(\fbox{x = 9, y = 12}\)
Step-by-step explanation:
\(\textsf {Let's solve by substitution.}\)
\(\rightarrow \mathsf {-5x + 4y = 3}\\\rightarrow \mathsf {x = 2y - 15}\)
\(\textsf {Substitute for x in the first equation of the system.}\)
\(\implies \mathsf {-5(2y-15)+4y=3}\)
\(\implies \mathsf {-10y+75+4y=3}\)
\(\implies \mathsf {-6y=-72}\)
\(\implies \textbf {y = 12}\)
\(\implies \mathsf {x = 2(12) - 15}\)
\(\implies \mathbf {x = 9}\)
\(\textsf {The solution is : x = 9, y = 12}\)
solve:
15.2+(-7.12)
NEEDED ASAP!
Answers
mark me brainliest if the answer is correct, thank you in advance :)
Am I the only person that is having problems with seeing answers?
Answers
Answer:
Been having issues all night
Answer:
Step-by-step explanation: no
What is the intermediate step in the form (x + a)2 = b as a result of completing the
square for the following equation?
x2 – 14 = 2x – 14
Answers
Answer:
y=(x-1)^2 -1
Step-by-step explanation:
If you put it in the form (x-a)^2=b
The answer will be (x-1)^2=1
