The general solution of differential equation is, y = k * (x²-9).
We can begin by separating the variables of the differential equation:
y′ = (2y) / (x²-9)
y′ / y = 2 / (x²-9)
Now we can integrate both sides with respect to their respective variables:
[tex]\int \dfrac{y'}{y} dy = \int \dfrac{2}{x^2-9} dx[/tex]
ln|y| = ln|x²-9| + C
where C is the constant of integration.
Simplifying:
|y| = e^(ln|x²-9|+C) = e^C * |x²-9|
Since e^C is a positive constant, we can write:
y = k * (x²-9)
where k is a non-zero constant. Therefore, the general solution of the given differential equation is y = k(x²-9), where k is any non-zero constant.
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--The complete question is, Find the general solution of the given differential equation. y′ = (2y) / (x²-9).--
An A.P has common difference d.If the sum of of the first twenty terms is twenty five times the first term, find in terms of d, the sum of thirty terms.
The sum of the first 30 terms in terms of d is 815d.
What is sum?In mathematics, the sum refers to the result of adding two or more numbers together. The process of adding numbers is called addition and the result of the addition is the sum.
What is arithmetic progression?An arithmetic progression (AP) is a sequence of numbers in which each term (except the first term) is obtained by adding a fixed constant to the preceding term. This fixed constant is called the common difference of the arithmetic progression.
According to given information:The sum of the first n terms of an arithmetic progression (A.P) is given by the formula:
[tex]S_n = [n/2] * [2a + (n-1)d][/tex]
where a is the first term and d is the common difference.
Given that the sum of the first 20 terms is 25 times the first term, we have:
[tex]S_{20} = 25a[/tex]
Substituting into the formula above, we get:
[tex]25a = [20/2] * [2a + (20-1)d]\\\\25a = 10a + 190d\\\\15a = 190d\\\\a = (190/15)d\\\\a = 38/3 d[/tex]
So the first term in terms of d is 38/3d.
Now we can use the formula to find the sum of the first 30 terms:
[tex]S_{30} = [30/2] * [2(38/3d) + (30-1)d]\\\\S_{30} = 15 * [76/3d + 29d]\\\\S_{30} = 5 * [76d + 87d]\\\\S_{30} = 815d[/tex]
Therefore, the sum of the first 30 terms in terms of d is 815d.
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Suppose we have a function defined by S x² – 6 f(x) = for x < 0, for x > 0. 10 - What values of a give f(x) = 43? Select the correct answer below: O x = -7,2 = 7. 2 = -7, x = 7, x = -33. a x = -7,2 = -33. O x= -7
Correct answer for function f(x) is: x = -√(43/S) for x < 0, and x = -11/2 for x > 0.
How to find the values of x that give f(x) = 43?We need to analyze the function separately for the cases x < 0 and x > 0.
1. For x < 0, the function is defined as f(x) = Sx². We need to find x such that Sx² = 43.
Sx² = 43
x² = 43/S
Since x < 0, we have x = -√(43/S)
2. For x > 0, the function is defined as f(x) = 10 - 6x. We need to find x such that 10 - 6x = 43.
10 - 6x = 43
-6x = 33
x = -33/6
x = -11/2
Thus, the correct answer is: x = -√(43/S) for x < 0, and x = -11/2 for x > 0.
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Tom buys a radio for £40
Later he sells it and makes a profit of 20%
Tom says:
"The ratio of the price I paid for the radio to the price I sold the radio is 5:6”
Enter a ratio that, when simplified, would show that Tom is correct.
Answer: he is correct
Step-by-step explanation:
40 x 1.2 = 48
40:48
divided by 8
=5.6
find the area under the standard normal curve to the right of z=0.81z=0.81. round your answer to four decimal places, if necessary
The area under the standard normal curve to the right of z=0.81 is approximately 0.2090. To find this area, we first look up the area to the left of z=0.81 in a standard normal table or calculator, which is approximately 0.7910. We then subtract this value from 1 since the total area under the standard normal curve is 1. The result is approximately 0.2090, which is the area under the standard normal curve to the right of z=0.81.
To find the area under the standard normal curve to the right of z=0.81, follow these steps:
1. Look up the z-score of 0.81 in a standard normal table or use a calculator with a built-in z-table function. This will give you the area to the left of z=0.81.
2. Since the total area under the standard normal curve is equal to 1, subtract the area to the left of z=0.81 from 1 to find the area to the right of z=0.81.
3. Round your answer to four decimal places, if necessary.
After looking up the z-score of 0.81 in a standard normal table, we find the area to the left is approximately 0.7910. Subtracting this value from 1, we get:
1 - 0.7910 = 0.2090
So, the area under the standard normal curve to the right of z=0.81 is approximately 0.2090.
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find f. f ''(x) = 8 cos(x), f(0) = −1, f(7/2) = 0
The final answr is F(x) = -8 cos(x) + (8cos(7/2)/7)x - 1.
Integrating, also known as integration, is a fundamental concept in calculus that involves finding the area under a curve or the accumulation of a quantity over a given interval. Integration is the opposite of differentiation, which involves finding the slope of a curve at a given point.
There are two main types of integrals: definite integrals and indefinite integrals. A definite integral involves finding the area under a curve over a specific interval, while an indefinite integral involves finding a function whose derivative is equal to the original function.
To find f given that f''(x) = 8 cos(x), we need to integrate this expression twice with respect to x to obtain f(x).
Integrating f''(x) once gives:
f'(x) = ∫ f''(x) dx = ∫ 8 cos(x) dx = 8 sin(x) + C1
where C1 is the constant of integration.
Integrating f'(x) once more gives:
f(x) = ∫ f'(x) dx = ∫ (8 sin(x) + C1) dx = -8 cos(x) + C1x + C2
where C2 is another constant of integration.
We can solve for the constants of integration using the initial conditions:
f(0) = -1 implies -8cos(0) + C1(0) + C2 = -1, so C2 = -1
f(7/2) = 0 implies -8cos(7/2) + C1(7/2) - 1 = 0, so C1 = 8cos(7/2)/7
Thus, the solution for f(x) is:
f(x) = -8 cos(x) + (8cos(7/2)/7)x - 1
Therefore, f(x) = -8 cos(x) + (8cos(7/2)/7)x - 1.
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Let f(x) = ax + b and g(x) = cx^2 + dx, where a, b, c, and d are constants. Compute f ◦ g and g ◦ f. Determine for which constants a, b, c, and d it is true that f ◦ g = g ◦ f. (Hint: Note that polynomials dnx n + dn−1x n−1 + · · · + d1x + d0 and enx n + en−1x n−1 + · · · + e1x + e0 are equal as functions if and only if dn = en, dn−1 = en−1, . . . , d1 = e1, d0 = e0.)
f(x) = x + b and g(x) =[tex]cx^2[/tex] are the only functions that satisfy [tex]f \circ g = g \circ f[/tex] for all constants b and c.
How to compute [tex]f \circ g[/tex]?To compute composition of function [tex]f \circ g[/tex], we substitute g(x) into f(x) and simplify:
f(g(x)) = a([tex]cx^2[/tex] + dx) + b
= [tex]acx^2[/tex]+ adx + b
To compute [tex]g \circ f[/tex], we substitute f(x) into g(x) and simplify:
[tex]g(f(x)) = c(ax + b)^2 + d(ax + b)[/tex]
[tex]= c(a^2x^2 + 2abx + b^2) + dax + db[/tex]
[tex]= ca^2x^2 + (2abc + da)x + cb^2 + db[/tex]
To find conditions under which [tex]f \circ g = g \circ f[/tex], we equate the expressions for[tex]f \circ g[/tex] and [tex]g \circ f[/tex] and simplify:
[tex]acx^2 + adx + b = ca^2x^2 + (2abc + da)x + cb^2 + db[/tex]
This is true for all x if and only if the coefficients of each power of x on both sides of the equation are equal. That is:
[tex]ac = ca^2, ad = 2abc + da, b = cb^2 + db[/tex]
Solving for a, b, c, and d, we get:
a = 0 or 1, b = 0, c = 0 or 1, d = 0
Therefore, f(x) = x + b and g(x) =[tex]cx^2[/tex] are the only functions that satisfy [tex]f \circ g = g \circ f[/tex] for all constants b and c.
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PLEASE HELP ME
The figure below shows roads near a pond. Each segment of the triangle represents a road or a path, except AB, which represents the distance across the pond.
Are the two triangles similar?
Yes the two triangles ΔCDE & ΔABC are similar according to the rules of similarity of triangles.
What is similarity?
If two triangles have the same proportion of matching sides to matching angles, they are said to be similar. Similar figures are items that share the same shape but differ in size between two or more figures or shapes.
Given that in ΔCDE,
∠DEC=55°
EC=40 ft
DE=25 ft
Also Given that in ΔCAB,
∠ABC=55°
BE=60 ft
Consider ΔCDE & ΔCAB
∠ABC = ∠DEC = 55°
∠C = ∠C
∠CAB =180-( ∠C+∠B)
=180-(∠C +55)
∠CDE= 180- (∠C+∠E)
=180-(∠C +55)
∠CAB =∠CDE=180-(∠C +55)
As three angles are congruent, the triangles are similar.
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solve differential equation dy/dx=y^2 . 16y(2)=0
The particular solution corresponding to the initial condition 16y(2) = 0 (which I assume means y(2) = 0), we can plug x = 2 and y = 0 into the equation:
-1/0 = 2 + C
To solve the differential equation dy/dx=y^2, we can separate the variables and integrate both sides.
dy/y^2 = dx
Integrating both sides:
-1/y = x + C
where C is the constant of integration. Solving for y:
y = -1/(x+C)
To solve the second part of the question, 16y(2) = 0, we substitute y(2) into the equation we just found:
y(2) = -1/(2+C)
16y(2) = 16*(-1/(2+C)) = -16/(2+C) = 0
Solving for C:
-16 = 0*(2+C)
Thus, C can be any value since 0 multiplied by any number is 0. Therefore, the solution to the differential equation dy/dx=y^2 and the equation 16y(2)=0 is y = -1/(x+ C), where C is any constant.
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A home has a rectangular kitchen. If listed as ordered pairs, the corners of the kitchen are (11, 5), (−6, 5), (11, −2), and (−6, −2). What is the area of the kitchen in square feet?
A. 119 ft2
B. 49 ft2
C. 48 ft2
Answer:
A. 119 ft2
Step-by-step explanation:
(11, 5) and (-6, 5)
= 11 - (-6)
= 17 feet
(11, 5) and (11, -2)
= 5 - (-2)
= 7 feet
17 × 7 = 119 square feet
halp il give all the points just help me
Answer:
Step-by-step explanation:
-5/2
To find the slope you need to use rise/run which is basically difference of y coordinates over difference of x coordinates
so first, pick 2 coordinates that you know in that linear relationship like in this case
(0,3) and (2,-2)
do rise/run which will look like this
=(y2-y1)/(x2-x1)
=(3-(-2))/(0-2)
=-5/2
suppose an = 2n2 n -4 .. find a closed formula for the sequence of differences by computing . simplify your answer as much as possible.
The closed formula for the sequence of differences is:
Δan = 3n
To find the sequence of differences for the given sequence, we subtract each term from the next term. So, the sequence of differences is:
2(2n + 1)
To find a closed formula for this sequence of differences, we can use the formula for the sum of the first n natural numbers:
sum = n(n+1)/2
Using this formula, we can write the sequence of differences as:
sum from i=1 to n of [2(2i + 1)]
= 2 sum from i=1 to n of [2i + 1]
= 2 [n(n+1) + n]
= 2n^2 + 4n
Therefore, the closed formula for the sequence of differences is 2n^2 + 4n.
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Find the distance from (-2,5) to (5,9) (round to the nearest tenth)
Answer:
8.1 hope this helps
Step-by-step explanation:
7 to the power of 2 and 4 to the power of 2
16 + 49 = 65
65 rounded to the nearest tenth is 8.1
Answer:
8.1
Step-by-step explanation:
Distance (d) = √(5 - -2)2 + (9 - 5)2
= √(7)2 + (4)2
= √65
= 8.0622577482985
After rounding
8.1
A train travelled along a track in 120 minutes, correct to the nearest 5 minutes
Sue finds out that the track is 290 km long.
She assumes that the track has been measured correct to the nearest 10 km.
a) Could the average speed of the train have been greater than 145 km/h? You must show how you get your answer and your final line must clearly say, 'Yes' or 'No'.
Sue's assumption was wrong.
The track was measured correct to the nearest 5 km.
b) What will the new maximum average speed be in km per minute? Give your answer correct to 2 decimal places.
Correct Answer gets brainliest.
in each of problems 4 through 6, find the laplace transform of the given function. 4. f (t) = t 0 (t − τ ) 2 cos(2τ ) dτ
The Laplace transform of the given function is:
L{f(t)} = (6 - 4s/(s²+4) + 2s²/(s²+4)²) / s⁴
To find the Laplace transform of the given function:
f(t) = t∫0 (t-τ)² cos(2τ) dτ
We will first factor out the constants outside the integral and write the function as:
f(t) = t ∫0 (t² - 2tτ + τ² ) cos(2τ) dτ
We can then break the integral into three parts and take the Laplace transform of each part separately, using the properties of the Laplace transform:
L{t} = 1/s²
L{t²} = 2/s³
L{cos(2τ)} = s/(s² + 4)
Using these Laplace transforms, we can write the Laplace transform of the given function as:
L{f(t)} = L{t ∫0 (t²- 2tτ + τ²) cos(2τ) dτ}
= L{t³} - 2L{t²}L{∫0 τ cos(2τ) dτ} + L{t}L{∫0 τ²cos(2τ) dτ}
= 6/s⁴ - 4/s⁴ * (s/(s²+4)) + 2/s⁴ * (s²(s²+4)² )
Simplifying this expression, we get:
L{f(t)} = (6 - 4s/(s²+4) + 2s²/(s²+4)²) / s⁴
Therefore, the Laplace transform of the given function is: L{f(t)} = (6 - 4s/(s²+4) + 2s²/(s²+4)²) / s⁴
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fill in the table using the function rule. y=19-2x
Using the function rule, y = 19 - 2x, the table can be filled as follows:
x y
1 17
3 13
4 11
6 7.
What is a function?A function is a mathematical equation that represents the relationship between the independent variable and the dependent variable.
The independent variable is the domain while the dependent variable is the codomain of the function.
The codomain depends on the domain.
x y
1 17 (19 -2(1)
3 13 (19 -2(3)
4 11 (19 -2(4)
6 7 (19 -2(6)
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A histogram of the sale price of (a subset of) homes in Ames, and a scatterplot of first floor area vs. sale price of the same homes are given below. 400 300 6e+05 200 4e+05 count Sale Price (dollars) 100 - 2e+05 Oe+00 - Oe+00 2e+05 8e+C 1000 3000 4e+05 6e+05 Sale Price (dollars) 2000 First Floor Area (sq. feet) (a) Describe the shape of the histogram of sale price of houses. (Where are the majority of sale prices located? Where are the minority of sale prices located?) (b) Are exponential, normal, or gamma distributions reasonable as the population distribution for the sale price of homes? Justify your answer. (c) Describe the relationship between first floor sq footage and sale price. (What happens to price as the area increases? What happens to the variability as area increases?)
The histogram of the sale price of houses appears to be skewed to the right, indicating that the majority of sale prices are located on the lower end of the price range. The majority of sale prices seem to be located between $100,000 and $400,000, with very few sale prices above $600,000.
An exponential distribution would not be a reasonable fit for the sale price of homes because it assumes a continuous variable with a constant rate of change. The sale price of homes is not a continuous variable, as it is determined by factors such as location, condition, and size. A normal distribution could potentially be a reasonable fit if the data was centered around a mean and did not have any significant outliers. However, as the histogram shows a skewed distribution, a gamma distribution may be a more appropriate fit as it allows for skewness in the data.
The scatterplot of first floor area vs. sale price shows a positive relationship between the two variables. As the first floor area increases, the sale price tends to increase as well. However, there appears to be a lot of variability in the sale price as the area increases. This suggests that other factors may be influencing the sale price of homes, in addition to the size of the first floor area.
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assume that all the given functions have continuous second-order partial derivatives. if z = f(x, y), where x = r2 s2 and y = 6rs, find ∂2z/(∂r ∂s).
Given functions have continuous second-order partial derivatives. Value of ∂²z/∂s∂r is 24r² + 48rs.
How to calculate ∂²z/∂s∂r?We can use the chain rule and product rule to find the second-order partial derivative of z with respect to r and s:
∂z/∂r = ∂z/∂x * ∂x/∂r + ∂z/∂y * ∂y/∂r
= 2rs * 2rs + 6s * 6r
= 24r²s + 24rs²
∂z/∂s = ∂z/∂x * ∂x/∂s + ∂z/∂y * ∂y/∂s
= 2rs * 2rs + 6r * 6s
= 24r²s + 36rs²
Taking the partial derivative of the first equation with respect to s, we get:
∂²z/∂s∂r = 24r² + 48rs
So the value of ∂²z/∂s∂r is 24r² + 48rs.
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If S is a subspace of R3 containing only the zerovector, what is Sperp?If S is spanned by (1,1,1), what is Sperp?If S is spanned by (2,0,0) and (0,0,3), what isSperp?I'm fairly sure that two vectors are orthogonal if their dotproduct is 0, but I want to make sure I'm doing this correctly.
1. f S is a subspace of R3 containing only the zerovector, the Sperp is equal to R3
2. If S is spanned by (1,1,1), the Sperp is spanned by (-1,-1,-2)
3. If S is spanned by (2,0,0) and (0,0,3), Sperp is spanned by (0,-6,0)
To find Sperp, we need to find the set of all vectors that are perpendicular to every vector in S.
1. If S only contains the zero vector, then any vector in R3 is perpendicular to every vector in S. Therefore, Sperp = R3.
2. If S is spanned by (1,1,1), then any vector that is orthogonal to (1,1,1) will be in Sperp. We can find such a vector by taking the cross product of (1,1,1) with any vector that is not parallel to it, say (1,-1,0):
(1,1,1) x (1,-1,0) = (-1,-1,-2)
So, Sperp is spanned by (-1,-1,-2).
3. If S is spanned by (2,0,0) and (0,0,3), then any vector that is orthogonal to both (2,0,0) and (0,0,3) will be in Sperp. We can find such a vector by taking the cross-product of the two spanning vectors:
(2,0,0) x (0,0,3) = (0,-6,0)
So, Sperp is spanned by (0,-6,0).
Note that in all cases, Sperp is a subspace of R3.
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A= 5 0 0 09 1 -3 4-4 -1 -2 1-4 -1 -7 6has two distinct real eigenvalues λ1<λ2. find the eigenvalues and a basis for each eigenspace. the smaller eigenvalue λ1 is_____ and a basis for its associated eigenspace is___ The larger eigenvalue λ2 is____ and a basis for its associated eigenspace is ____
The smaller eigenvalue λ1 is -2 and a basis for its associated eigenspace is {-1, 2, -1, 0}. The larger eigenvalue λ2 is 3 and a basis for its associated eigenspace is {0, -1, -1, 1}.
How to find the eigenvalues and eigenvectors?We need to solve the characteristic equation and the corresponding eigenvector equations.
The characteristic equation is:
det(A - λI) = 0
where I is the 4x4 identity matrix.
Expanding the determinant, we get:
(5 - λ)((1 - λ)(-7 - λ) - 6) - 0 + 0 - 0 = 0
Simplifying and solving for λ, we get:
λ^2 - λ - 6 = 0
(λ - 3)(λ + 2) = 0
So, the eigenvalues are λ1 = -2 and λ2 = 3.
Now, we need to find the eigenvectors corresponding to each eigenvalue.
For λ1 = -2, we need to solve the equation:
(A - λ1I)x = 0
Substituting λ1 = -2 and solving the system of equations, we get:
x1 = -1, x2 = 2, x3 = -1, x4 = 0
So, a basis for the eigenspace associated with λ1 is:
{-1, 2, -1, 0}
For λ2 = 3, we need to solve the equation:
(A - λ2I)x = 0
Substituting λ2 = 3 and solving the system of equations, we get:
x1 = 0, x2 = -1, x3 = -1, x4 = 1
Basis for the eigenspace connected to λ2 is:
{0, -1, -1, 1}
Therefore, the smaller eigenvalue λ1 is -2 and a basis for its associated eigenspace is {-1, 2, -1, 0}. The larger eigenvalue λ2 is 3 and a basis for its associated eigenspace is {0, -1, -1, 1}.
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21. An office has 6 floors. There are 148 employees on each floor. how many employees does the office have?
Answer:
888 employees
Step-by-step explanation:
We Know
An office has 6 floors.
There are 148 employees on each floor.
How many employees does the office have?
We Take
148 x 6 = 888 employees
So, the office has 888 employees.
Need help with logic puzzle ASAP
Need done my end of period 3:40pm
The preceding is a logic puzzle. Logic problems test the intellect and improve critical thinking.
The conclusions based on the clues providedJane was observed checking out an action book after leaving either a Biology or a History class, according to the indications. It was also discovered that Jayson is enrolled in Biology, and Jose, the kid who checked out a fantasy book, has an English class right after Jenny's.
Furthermore, we deduced that the person who left a History class was the same person who checked out a mystery novel, but the student studying French had to be present during 1st period.
Jaden, who is presently enrolled in Algebra, may be seen reading a Manga novel. It should be mentioned that while studying for academic topics such as Math, the urge to diverge into pleasure reading material can sometimes serve as a distraction.
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Working alone John can wash the windows of a building in 2.5 hours Caroline can wash the building windows by her self in 4 hours if they work together how many hours should it take to wash the windows
It should take John and Caroline approximately 0.1538 hours, or about 9.2 minutes, to wash the building windows when working together.
To solve this problem, we can use the formula:
Time taken when working together = (product of individual times) / (sum of individual times)
Let's first find the individual rates of work for John and Caroline:
John's rate of work = 1/2.5 = 0.4 windows per hour
Caroline's rate of work = 1/4 = 0.25 windows per hour
Now, we can substitute these values into the formula to find the time taken when working together:
Time taken = (0.4 x 0.25) / (0.4 + 0.25)
= 0.1 / 0.65
= 0.1538 hours
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if 3/4 cup of flour is used to make 4 individual pot pies, how much flour should be used to make 12 pot pies
Using proportion, amount of flour used to make 12 pot pies is 2.25 cups.
Given that,
Amount of flour used to make 4 individual pot pies = 3/4 cups
We have to find the amount of flour used to make 12 individual pot pies.
This can be found using the concept of proportion.
Using the concept of proportion,
Amount of flour used to make 1 individual pot pie = 3/4 ÷ 4
= 3/16 cups
Amount of flour used to make 12 individual pot pies = 12 × 3/16 cups
= 2.25 cups.
Hence the amount of flour used is 2.25 cups.
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Which action is an example of a medium-term savings goal?
A. Saving to buy a house
B. Saving to buy concert tickets
C. Saving to make a down payment on a used car
D. Saving for a new smartphone
Find the sum of the following series. Round to the nearest hundredth if necessary.
The sum of the finite geometric series in the problem is given as follows:
26,240.
How to obtain the sum of the finite geometric series?The first term of the series is given as follows:
[tex]a_1 = 8[/tex]
The common ratio of the series is given as follows:
r = 3.
(as each term is the previous term multiplied by 3).
The rule for the nth term of the series is given as follows:
[tex]a_n = 8(3)^{n - 1}[/tex]
Considering that the final term is of 17496, the value of n is given as follows:
[tex]17496 = 8(3)^{n - 1}[/tex]
3^(n - 1) = 2187
3^(n - 1) = 3^7
n - 1 = 7
n = 8.
Hence the sum of the series is given as follows:
S = [8 - 8 x 3^8]/-2
S = 26,240.
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Jerry’s grandmother worked in a department store for many years. Now that she has retired,she receives a monthly Social Security check.Jerry’s grandmother and her employer paid a tax during her working years that helped fund Social Security. Which is the tax?
57 .99 rounded to two decimals places
y varies directly as a square of z and inversely as x. if the constant variation is -2. what is the equation that relates y,x, and z
The equation that relates y,x, and z when constant variation is -2: y = -2 *(z²) / x.
What is equation?An equation is a mathematical statement that shows the equality of two expressions. It usually consists of two sides separated by an equal sign (=). The expressions on both sides of the equal sign can include numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. An equation can have one or more unknown variables, and the goal is often to find the values of these variables that make the equation true. Equations are used in many different areas of mathematics and science to describe relationships between quantities, to solve problems, and to model real-world phenomena.
Here,
If y varies directly as the square of z and inversely as x, we can write:
y = k * (z²) / x
where k is the constant of variation. We are told that the constant of variation is -2, so we can substitute this value into the equation:
y = -2 * (z²) / x
This is the equation that relates y, x, and z.
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What is coefficeient of x^9 in (2-x)^19?
The coefficient of x⁹ in (2-x)¹⁹ is -48620.
To find the coefficient of x⁹ in (2-x)¹⁹, we use the binomial theorem. The general term of a binomial expansion is given by:
T(r+1) = nCr * [tex]a^(^n^-^r^)[/tex] * [tex]b^r[/tex]
where n is the power (19 in this case), r is the term index, a is the first term (2), b is the second term (-x), and nCr represents the binomial coefficient.
For the x⁹ term, we need to find T(9+1) or T(10). Plugging in the values, we get:
T(10) = 19C9 * 2⁽¹⁹⁻⁹⁾ * (-x)⁹
T(10) = 19C9 * 2¹⁰ * (-1)⁹ * x⁹
19C9 can be calculated as 19! / (9! * 10!) = 92378.
So, T(10) = 92378 * 2¹⁰ * (-1)⁹ * x⁹ = -48620 * x⁹.
Hence, the coefficient of x⁹ in (2-x)¹⁹ is -48620.
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(1 point) find a particular solution to ″ 6′ 8=54.
Therefore, a particular solution to the equation y″ + 6y′ + 8y = 54 is yp = 27/4.
To find a particular solution to the equation y″ + 6y′ + 8y = 54, we can use the method of undetermined coefficients.
First, identify the general form of the particular solution based on the non-homogeneous term: Since the right side of the equation is a constant (54), we can guess that the particular solution will be in the form of yp = A, where A is a constant.
Next, substitute the guess into the equation: The first and second derivatives of yp = A are both 0 (y′ = 0, y″ = 0). So, substituting into the equation, we get 0 + 6(0) + 8A = 54.
Now, solve for the constant A: 8A = 54, so A = 54/8 = 27/4.
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