For W1, W2⊂W1, W2⊂V be finite-dimensional subspaces of a vector space V,
(a) If either W1 or W2 is the zero subspace, the formula holds.
(b) Given a basis S for the intersection W1∩W2, there exist sets T1 and T2 such that their union with S forms bases for W1 and W2, respectively.
(c) The union U = S∪T1∪T2 spans the sum W1 + W2.
(d) The union U = S∪T1∪T2 is linearly independent, and the formula dim(W1 + W2) = dim(W1) + dim(W2) - dim(W1∩W2) holds.
(a) If W1 = {0}, then the dimension of W1 is 0. Similarly, if W2 = {0}, then the dimension of W2 is 0. In both cases, the intersection of W1 and W2, denoted by W1∩W2, is also {0}, and its dimension is 0.
Therefore, we have:
dim(W1 + W2) = dim({0} + W2) = dim(W2) = dim(W1) + dim(W2) - dim(W1∩W2) = 0 + dim(W2) - 0 = dim(W2).
(b) Let S be a basis of W1∩W2. Since W1 and W2 are nonzero, they each have at least one nonzero vector. Let v1 be a nonzero vector in W1, and v2 be a nonzero vector in W2. Then {v1} is linearly independent and can be extended to a basis T1 of W1. Similarly, {v2} is linearly independent and can be extended to a basis T2 of W2.
(c) To show that U = S∪T1∪T2 spans W1 + W2, we need to show that every vector in W1 + W2 can be expressed as a linear combination of vectors in U.
Let w be an arbitrary vector in W1 + W2. By definition, there exist vectors w1 ∈ W1 and w2 ∈ W2 such that w = w1 + w2. Since T1 is a basis of W1, we can express w1 as a linear combination of vectors in T1. Similarly, w2 can be expressed as a linear combination of vectors in T2. Therefore, we can write w as a linear combination of vectors in U = S∪T1∪T2.
(d) To show that U = S∪T1∪T2 is linearly independent, we need to show that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where ci are scalars and vi are vectors in U, is the trivial solution c1 = c2 = ... = cn = 0.
Since S is a basis of W1∩W2, any vector in S can be expressed as a linear combination of vectors in S. Similarly, vectors in T1 can be expressed as a linear combination of vectors in T1, and vectors in T2 can be expressed as a linear combination of vectors in T2. Therefore, the equation c1v1 + c2v2 + ... + cnvn = 0 implies that the coefficients ci must be zero for all vectors in U.
By showing that U is linearly independent, we can deduce that dim(W1 + W2) = |U| = |S∪T1∪T2| = |S| + |T1| + |T2| = dim(W1) + dim(W2) - dim(W1∩W2), which is the claimed identity.
Therefore, we have proved that dimension dim(W1 + W2) = dim(W1) + dim(W2) - dim(W1∩W2).
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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 ~ 233.7
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 about 37.2. Plug 37.2 into the formula 2(pi)r to find the Circumference. The answer is 233.7 rounded to the nearest tenth.
A triangle has three angles that measure 50 degrees, 28 degrees and 3x. What is the value of x?
Answer:
57.33
Step-by-step explanation:
50+28+3x=180
78+3x=180-78
3x=172/3
x = 57.33
What is another way to write -8+5?
Answer:
5-8
Step-by-step explanation:
you just had to move the 8 over
Which angles are neither obtuse angles nor acute angles?
Answer:
Acute angles are less than 90
obtuse is more than 90
right angles are exactly 90
Step-by-step explanation:
Answer:
First option: 90 degrees
Step-by-step explanation:
Obtuse is greater than 90.
Acute is less than 90.
Right is 90.
There are 35 green, 22 white, 30 purple and 14 blue gumballs in the gumball machine.
Sharee wants to get a white or green gumball.
What is the probability of getting a white or green gumball?
Answer:
The probability of getting a white or green gumball is 35.64%.
Step-by-step explanation:
1. calculate the total number of gumballs in the gumball machine
35 + 22 + 30 + 14 = 101 gumballs
2. calculate the number of white and green gumballs combined
22 + 14 = 36 w/g gumballs
3. divide the number of white and green gumballs combined by the total number of gumballs in the gumball machine
36 / 101 = 0.3564
Answer: the probability of getting a white or green gumball is 35.64% (0.3564).
Probability is the chance of occurring an event from the total possible outcomes.
The probability of getting a white or a green gumball is 57/101.
What is probability?It is the chance of occurring an event from the total possible outcomes.
We have,
Green gumballs = 35
White gumballs = 22
Purple gumballs = 30
Blue gumballs = 14
Total gumballs = 35 + 22 + 30 + 14 = 101
The combination is given by:
= [tex]^nC_{r}[/tex]
= n! / r! (n-r)!
The probability of choosing a white gumball:
= [tex]\frac{^{22}C_{1}}{^{101}C_{1} }[/tex]
= 22 / 101
The probability of choosing a green gumball:
= [tex]\frac{^{35}C_{1}}{^{101}C_{1} }[/tex]
= 35 / 101
The probability of getting a white or a green gumball:
= 22/101 + 35 / 101
= (22 +35) / 101
= 57 / 101
Thus the probability of getting a white or a green gumball is 57/101.
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4. Use polynomial fitting to find the closed form for the sequence 2, 5, 11, 21, 36, ...
The sequence continues as follows:2, 5, 11, 21, 36, 67, ...
Given sequence2 5 11 21 36...
First differences 3 6 10 15...
Second differences 3 4 5...
Third differences 1 1...
The third differences are constant, which means that we can use a cubic polynomial for the fitting.
The formula for a cubic polynomial is
an³ + b
n² + c
n + d
Let us denote the nth term of the sequence by fn. Then, we have
f1 = 2, f2 = 5, f3 = 11, f4 = 21, f5 = 36...
We can write a system of equations using the first four terms of the sequence.
2 = a + b + c + d
5 = 8a + 4b + 2c + d
11 = 27a + 9b + 3c + d
21 = 64a + 16b + 4c + d
Solving this system, we get a = 1/3, b = 1, c = 11/3, and d = 0.
Thus, the closed-form expression for the nth term of the sequence isf(n) = (1/3)n³ + n² + (11/3)n
The next term in the sequence is f(6) = (1/3)(6)³ + (6)² + (11/3)(6) = 67.
Therefore, the sequence continues as follows:2, 5, 11, 21, 36, 67, ...
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LA Galaxy have won 35% of there soccer matches, and drawn 9 of them. If they played 40 matches, how many have they lost?
Answer:17
Step-by-step explanation:
0.35•40=14 to find how many they’ve won
14+9=23 is how many they have won or drawn
40-23=17 is how many they’ve lost
Somebody help me please
Answer:
Other dude is correct
Step-by-step explanation:
Identify the coefficient in the following expression 6a +7
Answer:
The coefficient is 6
Step-by-step explanation:
Newton has just gone grocery shopping. The mean cost for each item in his bag was $3.38. He bought a total of 6 items, and the prices of 5 of those items are listed below:
$4.09, $4.03, S4.19, $2.24, $4.09
Determine the price of the 6th item in his bag.
Newton has just gone grocery shopping. The mean cost for each item in his bag was $3.38. The price of the 6th item in his bag is $1.64
To determine the price of the 6th item in Newton's bag, we can use the concept of the mean (average). We know that the mean cost for each item in his bag is $3.38, and he bought a total of 6 items.
To find the sum of all 6 item prices, we can multiply the mean cost by the total number of items:
Sum of all item prices = Mean cost * Total number of items
= $3.38 * 6
= $20.28
We also know the prices of 5 of the items, which are $4.09, $4.03, $4.19, $2.24, and $4.09.
To determine the price of the 6th item, we subtract the sum of the known prices from the sum of all item prices:
Price of the 6th item = Sum of all item prices - Sum of known prices
= $20.28 - ($4.09 + $4.03 + $4.19 + $2.24 + $4.09)
= $20.28 - $18.64
= $1.64
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Factor completely please. 3k^2-19k+20
Answer:
3(k*k)-3(3k)-5(2k)+2(5+5)
Step-by-step explanation:
3k²-19k+20
Just find some number that equal the terms when multiplied. ¯\_(ツ)_/¯
DOUBLE CHECK!
3(k*k)-3(3k)-5(2k)+2(5+5)
Multiply what you need to.
3k²-9k-10k+20
Combine like terms.
3k²-19k+20
---
hope it helps
For each of the following rejection regions, what is the probability that a Type I error will be made?
a. t> 2.718, where df = 11
b. t< -1.476, where df = 5
c. t< -2.060 or t > 2.060, where df = 25
a. The probability that a Type I error will be made is _____ (Round to two decimal places as needed.)
The probability of a Type I error in rejection region a is 0.01.
What is probability?We must find the area under the t-distribution curve outside the rejection region, assuming a two-tailed test in order to to determine the probability of making a Type I error in each rejection region.
for a. t > 2.718, and df = 11:
Using a t-distribution table, the probability is 0.01.
b. t < -1.476, df = 5:
Using a t-distribution table the probability is 0.05.
c. t < -2.060 or t > 2.060, df = 25:
Using a t-distribution table, the area in each tail is 0.025.
The combined probability of a Type I error in rejection region c is
0.025 + 0.025 = 0.05.
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solve the differential equation by variation of parameters. y'' y = csc(x)
The solution to the differential equation y'' + y = csc(x) using the variation of parameters method is y(x) = cos(x)ln|sin(x)| + Csin(x), where C is a constant.
To solve the differential equation by variation of parameters, we first find the complementary solution (the solution to the homogeneous equation). The homogeneous equation is y'' + y = 0, which has the solution y_c(x) = Acos(x) + Bsin(x), where A and B are constants.
Next, we find the particular solution using the variation of parameters method. We assume the particular solution is of the form y_p(x) = u(x)cos(x) + v(x)sin(x), where u(x) and v(x) are unknown functions.
We differentiate y_p(x) to find y_p' and y_p'' and substitute them into the original differential equation. After simplification, we obtain u'(x)sin(x) - v'(x)cos(x) = csc(x).
To solve this system of equations, we find the derivatives u'(x) and v'(x) and integrate them to obtain u(x) and v(x). Finally, we substitute u(x) and v(x) into the particular solution form y_p(x) = u(x)cos(x) + v(x)sin(x).
The final solution is y(x) = y_c(x) + y_p(x), which simplifies to y(x) = cos(x)ln|sin(x)| + Csin(x), where C is the constant of integration.
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Which graphed function is described by the given intervals? Increasing on (−∞, 0) Decreasing on (0, ∞)
A) A
B) B
C) C
D) D
Write a cosine function that has a midline of y=5, an amplitude of 3, a period of 1, and a horizontal shift of 1/4 to the right
The graph of this function begins at 5 and moves down to 2 and back to 5 again in a repeated pattern over one period of 1.
A cosine function is a periodic function that fluctuates about its midline and follows a predictable pattern. The formula for a cosine function with a midline of y = c, amplitude of a, and period of b is:y = a cos(bx) + cTo shift the graph of a cosine function,
you can add or subtract a value inside the parentheses of the formula, which results in a horizontal shift.
The shift is to the right if you add, and to the left if you subtract.In this instance,
the cosine function has the following characteristics:Midline = y = 5Amplitude = 3Period = 1Horizontal shift to the right = 1/4We'll have to adjust the formula to include all of these parameters.
First and foremost, let's figure out the function's frequency, which is determined by dividing 2π by the period of the cosine function. In this example, the frequency is 2π/1, which equals 2π.
y = a cos(bx) + c is the formula we'll use. We'll substitute the values given into the formula. The resulting formula is:y = 3cos(2π(x - 1/4)) + 5This is the cosine function with a midline of y = 5, an amplitude of 3, a period of 1, and a horizontal shift of 1/4 to the right.
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Solve the system with the addition method: - 8x + 5y = -33 +8.x – 4y = 28 Answer: (x, y) Preview 2 Preview y Enter your answers as integers or as reduced fraction(s) in the form A/B.
The solution to the system -8x + 5y = -33 and 8x - 4y = 28 is (x, y) = (7, -1).
To solve the given system of equations using the addition method, let's eliminate one variable by adding the two equations together. The system of equations is:
-8x + 5y = -33 (Equation 1)
8x - 4y = 28 (Equation 2)
When we add Equation 1 and Equation 2, the x terms cancel out:
(-8x + 5y) + (8x - 4y) = -33 + 28
y = -5
Now that we have the value of y, we can substitute it back into either Equation 1 or Equation 2 to solve for x. Let's use Equation 1:
-8x + 5(-5) = -33
-8x - 25 = -33
-8x = -33 + 25
-8x = -8
x = 1
Therefore, the solution to the system is (x, y) = (1, -5).
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In a simple random sample of 170 households, the sample mean number of personal computers was1.71. Assume the population standard deviation is σ=0.86.
(a) Construct a 90% confidence interval for the mean number of personal computers. Round the answer to at least two decimal places.
A 90% confidence interval for the mean number of personal computers is _______
(b) If the sample size were 120 rather that 170, would the margin of error be larger or smaller than the result in part (a)? Explain.
The margin of error would be _______, since __________in the sample size will _______ the standard error.
(c) If the confidence levels were 95% rather than 90%, would the margin of error be larger or smaller than the result in part (a)? Explain.
The margin of error would be _______, since __________ in the confidence level will ___________the critical value za/2.
(d) Based on the confidence interval constructed in part (a), is it likely that the mean number of personal computers is greater than 1?
It ________ likely that the mean number of personal computers is greater than1.
The confidence interval using a 90% confidence level is (1.601, 1.819)
Confidence IntervalConfidence Interval = Sample Mean ± Margin of Error
Given the parameters:
Sample Mean (x) = 1.71
Population Standard Deviation (σ) = 0.86
Sample Size (n) = 170
Confidence Level = 90%
The standard Error(SE) is given by:
SE = σ / √n
SE = 0.86 / √170 ≈ 0.0663
The margin of Error is related to standard error by the relation:
ME = Z × SE
ME = 1.645 × 0.0663 ≈ 0.109
Confidence Interval = Sample Mean ± Margin of Error
Confidence Interval = 1.71 ± 0.109
Confidence Interval= (1.601, 1.819)
Part B:A decrease in sample size from 170 to 120 would lead to increase in error margin , since decrease in sample size will increase the standard error.
Part C:If confidence interval were raised from 90% to 95% , the error margin would be larger since increase in confidence interval would increase the critical value
Part D :Based on the confidence interval , it is very likely that the mean number of personal computers is greater than 1.
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evaluate ∫30(4f(t)−6g(t)) dt given that ∫150f(t) dt=−7, ∫30f(t) dt=−8, ∫150g(t) dt=4, and ∫30g(t) dt=8
The evaluation of ∫30(4f(t) - 6g(t)) dt given that ∫150f(t) dt = -7, ∫30f(t) dt = -8, ∫150g(t) dt = 4, and ∫30g(t) dt = 8 is -80.
Given that ∫150f(t) dt = -7, ∫30f(t) dt = -8, ∫150g(t) dt = 4, and ∫30g(t) dt = 8.
Let us evaluate ∫30(4f(t) - 6g(t)) dt.
Therefore,∫30(4f(t) - 6g(t)) dt = ∫30(4f(t) dt - 6g(t) dt) = 4 ∫30f(t) dt - 6 ∫30g(t) dt
Now, using the given values in the question we can say that,∫30(4f(t) - 6g(t)) dt = 4 ∫30f(t) dt - 6 ∫30g(t) dt = 4 (-8) - 6(8) = -32 - 48 = -80
Therefore, the evaluation of ∫30(4f(t) - 6g(t)) dt given that ∫150f(t) dt = -7, ∫30f(t) dt = -8, ∫150g(t) dt = 4, and ∫30g(t) dt = 8 is -80.
Note: The given integrals ∫150f(t) dt, ∫30f(t) dt, ∫150g(t) dt, and ∫30g(t) dt are only intermediate steps in order to evaluate the final integral.
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what is the distance between (3,-5), (-3,0)
Answer: 11 units
Step-by-step explanation:
-3 and 3 are 6 units apart, so that would be 6 units, and when you add the -5 into the problem, the 6 units become 11 units. Hope this helped :)
Answer:
Exact Form:
√ 61
Decimal Form:
7.81024967 …
A triangular pane of glass has a height of 30 inches and an area of 240 square inches. What is the length of the base of the pane?
240÷30=8
8×30=240
30 height 8 with
If you have a circle with a central angle of 80 degrees, what is the degrees of its inscribed angle?
Answer:
The measure of the inscribed angle is [tex]40^{\circ}[/tex]
Step-by-step explanation:
Recall that the measure of the inscribed angle is one-half the measure of the central angle subtended by the same arc.
So, if the measure of the central angle is [tex]80^{\circ}[/tex], so the measure of the corresponding inscribed angle is:
[tex]\frac{1}{2}\times 80^{\circ}=40^{\circ}[/tex]
An object located at the point (4, 55) on a distance time graph is later located at the point (9, 45). If distance is in metres and time is in seconds, the average speed is: a) -2 m/s b) 5 m/s c) -5 m/s d) 2 m/s
An object located at the point (4, 55) on a distance time graph is later located at the point (9, 45). If distance is in metres and time is in seconds, the average speed is -2 m/s.
To calculate the average speed of an object, we need to find the total distance traveled divided by the total time taken.
Given that the object is located at the point (4, 55) initially and later located at the point (9, 45), we can determine the total distance traveled and the total time taken.
Total distance traveled = Difference in distance = 45 - 55 = -10 meters (negative because the object moved from a higher distance to a lower distance)
Total time taken = Difference in time = 9 - 4 = 5 seconds
Average speed = Total distance traveled / Total time taken = -10 meters / 5 seconds = -2 meters/second
Therefore, the average speed of the object is -2 m/s.
Option (a) -2 m/s is the correct answer.
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help me
The two major types of metal electrical cable are:
A.wire and fibre
B.twisted pair and coaxial
C.twisted pair and fibre optic
D.coaxial and fibre optic
2. What are the most common forms of wireless connection?
A.infrared beams
B.microwaves
C.radio waves
D.all of the above
3. Wi-fi connections have limited range of :
A.10 metres
B.600 metres
C.20 metres
D.300 metres
Answer:
1.The two major types of metal electrical cable are:
A.wire and fibre
2. What are the most common forms of wireless connection?
C.Radio news
3. Wi-fi connections have limited range of :
D.300 metres
Solve the system of differential equations s = Ly' - 29x + 42y – 20x + 294 = x(0) = – 15, y(0) = == - 11 The lesser of the two eigenvalues is Its corresponding eigevector is (a, – 2). What is a? a = The greater of the two eigenvalues is Its corresponding eigevector is ( – 7,6). What is b? b = = The solution to the system is x(t) = = y(t) =
The lesser of the two eigenvalues is λ = -1. Its corresponding eigenvector is [-42 29].
The greater of the two eigenvalues is λ = 1. Its corresponding eigenvector is [42 29].
To solve the system of differential equations:
x' = -29x + 42y
y' = -20x + 29y
We can rewrite it in matrix form as follows:
X' = AX
where X = [x y] is a vector, X' represents the derivative of X with respect to time, and A is the coefficient matrix. In this case, A is given by:
A = [ -29 42 ]
[ -20 29 ]
To find the eigenvalues and eigenvectors, we need to solve the characteristic equation:
det(A - λI) = 0
where λ is the eigenvalue and I is the identity matrix. Solving this equation will give us the eigenvalues, and by substituting these eigenvalues back into the equation (A - λI)V = 0, where V is the corresponding eigenvector, we can find the eigenvectors.
Let's calculate the eigenvalues first. We have:
A - λI = [ -29 42 ]
[ -20 29 ] - λ [ 1 0 ]
[ 0 1 ]
Expanding the determinant, we get:
(-29 - λ)(29 - λ) - (42)(-20) = 0
(λ + 29)(λ - 29) + 840 = 0
λ² - 29² + 840 = 0
λ² - 841 + 840 = 0
λ² = 1
λ = ±1
So the eigenvalues are λ = 1 and λ = -1.
To find the corresponding eigenvectors, we substitute each eigenvalue back into the equation (A - λI)V = 0.
For λ = 1, we have:
[ -29 42 ] [ v₁ ] [ 0 ]
[ -20 29 ] [ v₂ ] = [ 0 ]
This gives us the following system of equations:
-29v₁ + 42v₂ = 0
-20v₁ + 29v₂ = 0
Solving this system, we find that v₁ = 42 and v₂ = 29.
Therefore, the eigenvector corresponding to the eigenvalue λ = 1 is [42 29].
For λ = -1, we have:
[ -29 42 ] [ v₁ ] [ 0 ]
[ -20 29 ] [ v₂ ] = [ 0 ]
This gives us the following system of equations:
-29v₁ + 42v₂ = 0
-20v₁ + 29v₂ = 0
Solving this system, we find that v₁ = -42 and v₂ = 29.
Therefore, the eigenvector corresponding to the eigenvalue λ = -1 is [-42 29].
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Complete Question:
Solve the system of differential equations
x' = -29x + 42y
y' = -20x + 29y
x(0) = -15, y(0) = -11
The lesser of the two eigenvalues is Its corresponding eigenvector is
The greater of the two eigenvalues is Its corresponding eigenvector is
For the following exercise, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci. 9x² + 36y²-36x + 72y +36 = 0
The given equation, 9x² + 36y² - 36x + 72y + 36 = 0, represents an ellipse. In standard form, the equation can be written as (x-1)²/4 + (y+1)²/1 = 1. The center of the ellipse is at (1, -1), the vertices are located at (3, -1) and (-1, -1), and the foci are at (2, -1) and (0, -1).
To write the equation 9x² + 36y² - 36x + 72y + 36 = 0 in standard form, we need to complete the square for both the x and y terms. By rearranging the equation, we have 9x² - 36x + 36 + 36y² + 72y + 36 = 0.
Next, we can factor out a 9 from the x terms and a 36 from the y terms: 9(x² - 4x + 4) + 36(y² + 2y + 1) = 0.
Simplifying further, we have 9(x - 2)² + 36(y + 1)² = 36.
Dividing both sides by 36, we get (x - 2)²/4 + (y + 1)²/1 = 1, which is the standard form of an ellipse.
From the standard form, we can determine that the center of the ellipse is located at (1, -1), the vertices are at (3, -1) and (-1, -1), and the foci are at (2, -1) and (0, -1).
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The starting salary of a starting teacher is $40,000, which will increase by 2% each year
What is the multiplier (growth factor) for this scenario?
1. .02
2. 1.02
3. 2
4. 1.2
Answer:
Choice B - (1.02)
Step-by-step explanation:
With the teachers starting salary being $40,000, we can rightfully assume that it their salary would not increase with Choice A, $40,000 * 0.2 = $800.
2% otherwise known as 0.02, would make the teachers yearly salary $40,000 * 1 + 0.02, or $40,000 * 1.02. Making the correct answer, Choice B - (1.02). (This would also give the teacher an additional $800 per year!)
identify the slope and y-intercept of the lins given by the equation y=2x 1.
Simplify 4(8x).
A. 8x
B. 4x
C. 32x
D. 8 + 4x
Answer:
32x
Step-by-step explanation: 4(8x)=32x
How to do this with steps please help !
Answer:
126+111+43+8=288
360-288=72
72÷12=6
x=6
Answer:
x=6
Step-by-step explanation:
Step by step explaination attached below.
y=Ax^2 + C/x is the general solution of the DEQ: y' + y/x = 39x. Determine A. Is the DEQ separable, exact, 1st-order linear, Bernouli?
The exact value of A in the general solution is 13
Also, the DEQ is separable
How to determine the value of A in the general solutionFrom the question, we have the following parameters that can be used in our computation:
y = Ax² + C/x
The differential equation is given as
y' + y/x = 39x
When y = Ax² + C/x is differentiated, we have
y' = 2Ax - Cx⁻²
So, we have
2Ax - Cx⁻² + y/x = 39x
Recall that
y = Ax² + C/x
So, we have
2Ax - Cx⁻² + (Ax² + C/x)/x = 39x
Evaluate
2Ax - Cx⁻² + Ax + Cx⁻² = 39x
This gives
2Ax + Ax = 39x
So, we have
3Ax = 39x
By comparing both sides of the equation, we have
3A = 39
Divide both sides by 3
A = 13
Hence, the value of A in the general solution is 13
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