For the following homogeneous differential equation, given that y/₁(x) = ex is a solution, find the other independent solution y2. Then, check explicitly that y1 and y2 are independent.
(2 + x) d2y/dx2 – (2x + 3) dy/dx + (x+1) y= 0
Answers
The other independent solution y₂ for the given homogeneous differential equation is y₂(x) = e^(−x).
To find y₂, we start by assuming y₂(x) = e^(rx), where r is a constant to be determined. We then differentiate y₂ twice with respect to x and substitute these expressions into the differential equation:
(2 + x) * [d²(e^(rx))/dx²] - (2x + 3) * [d(e^(rx))/dx] + (x + 1) * e^(rx) = 0.
After simplification and collecting like terms, we get:
(2r² + 2r) * e^(rx) - (2rx + 3r) * e^(rx) + (x + 1) * e^(rx) = 0.
Since e^(rx) is nonzero for all x, we can divide the entire equation by e^(rx) to obtain:
2r² + 2r - 2rx - 3r + x + 1 = 0.
Rearranging the terms, we have:
2r² - (2x + 3) * r + (x + 1) = 0.
This equation must hold for all x, so the coefficients of each term must be zero. By comparing coefficients, we get the following system of equations:
2r² = 0,
2r - (2x + 3) = 0,
x + 1 = 0.
The first equation yields r = 0. Substituting this into the second equation, we find:
2 * 0 - (2x + 3) = 0,
-2x - 3 = 0,
x = -3/2.
However, this value does not satisfy the third equation, x + 1 = 0. Therefore, r = 0 does not yield a valid solution.
We need a different value for r that satisfies all three equations. Let's consider r = -1. Substituting this into the second equation, we get:
2 * (-1) - (2x + 3) = 0,
-2 - 2x - 3 = 0,
-2x - 5 = 0,
x = -5/2.
This value satisfies all three equations, so we can conclude that y₂(x) = e^(−x) is the other independent solution.
To check if y₁(x) = e^x and y₂(x) = e^(−x) are independent, we can evaluate their Wronskian determinant:
W[y₁, y₂](x) = |e^x e^(−x)| = e^x * e^(−x) - e^(−x) * e^x = 0.
Since the Wronskian determinant is zero for all x, we can conclude that y₁ and y₂ are dependent.\
To know more about coefficients, refer here:
https://brainly.com/question/1594145#
#SPJ11
Related Questions
The graph represents the distribution of the lengths of play times, in minutes, for songs played by a radio station over one hour.
A graph shows the horizontal axis numbered 2.6 to x. The vertical axis is unnumbered. The graph shows an upward trend from 2.8 to 3.4 then a downward trend from 3.4 to 4.
Which statement is true about the songs played during the one-hour interval?
Most of the songs were between 3 minutes and 3.8 minutes long.
Most of the songs were 3.4 minutes long.
Most of the songs were less than 3.2 minutes long.
Most of the songs were more than 3.6 minutes long.
Answers
The correct statement is Most of the songs were between 3 minutes and 3.8 minutes long.
Based on the given information from the graph, we can determine the following:
The graph shows an upward trend from 2.8 to 3.4 on the horizontal axis.
Then, there is a downward trend from 3.4 to 4 on the horizontal axis.
From this, we can conclude that most of the songs played during the one-hour interval were between 3 minutes and 3.8 minutes long. This is because the upward trend indicates an increase in length from 2.8 to 3.4, and the subsequent downward trend suggests a decrease in length from 3.4 to 4.
Therefore, the correct statement is:
Most of the songs were between 3 minutes and 3.8 minutes long.
for such more question on minutes
https://brainly.com/question/25279049
#SPJ8
Answer:
A
Step-by-step explanation:
Jay interpreted the meaning of the given data set, 7.76, as meaning half the couch weigh less than 7.76 pounds in weighed half more. What was his error? Explain.
Answers
Jay's error is that he misinterpreted the concept of the mean of the data set. The mean, also known as the average, is the sum of all numbers in a data set divided by the total number of data points. In this case, we do not know the context of the data set, so we cannot assume it refers to weight or the weight of a couch.
Jay's interpretation of the mean as "half the couch weight less than 7.76 pounds and half weighed more" is incorrect because it assumes that the data follows a symmetrical distribution, which may not be the case. The mean can be influenced by the values on either side of it, and it does not necessarily divide the data set in half.
If we assume that the data set refers to the weight of couches, then Jay's interpretation is incorrect because it is impossible for half the weight of a couch to be less than its weight. Rather, weight is always a positive value, so it would be more appropriate to describe the data in terms of a range or distribution rather than in terms of half and half.
In conclusion, Jay's error was in misinterpreting the meaning of the mean, assuming a symmetrical distribution, and making assumptions about the data set without the necessary context. It is important to understand the context and statistical properties of data before making any conclusions or interpretations.
Hopefully, I correctly answered your question! If I did, I would really appreciate Brainliest. You can give Brainliest by clicking the crown (it only works if there’s two people who answered.) I would also appreciate if you rated my answer 5 stars, and clicked the heart!
calculate the taylor polynomials 2 and 3 centered at = for the function ()=25ln( 1), =0. (express numbers in exact form. use symbolic notation and fractions where needed.)
Answers
Therefore, the Taylor polynomials T2(x) and T3(x) centered at a=0 for the function f(x) = 25ln(1+x) are:
T2(x) = 25 + 25x - 12.5x^2
T3(x) = 25 + 25x - 12.5x^2 + 16.67x^3
To calculate the Taylor polynomials T2(x) and T3(x) centered at a=0 for the function f(x) = 25ln(1+x), we need to find the derivatives of f(x) and evaluate them at x=0.
First, let's find the derivatives of f(x):
f'(x) = 25/(1+x)
f''(x) = -25/(1+x)^2
f'''(x) = 50/(1+x)^3
Now, let's evaluate these derivatives at x=0:
f'(0) = 25/(1+0) = 25
f''(0) = -25/(1+0)^2 = -25
f'''(0) = 50/(1+0)^3 = 50
Using these values, we can now write the Taylor polynomials:
T2(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 = 25 + 25x - 12.5x^2
T3(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 = 25 + 25x - 12.5x^2 + (16.67)x^3
To know more about Taylor polynomials,
https://brainly.com/question/30258832
#SPJ11
if a destination goal is created for a newsletter sign-up and a user completes the newsletter sign-up three times in three separate sessions, how many goal conversions will analytics count?3621
Answers
Analytics will count three goal conversions if a user completes the newsletter sign-up three times in three separate sessions.
When a goal is set up in analytics, it is assigned a unique ID. When a user completes the desired action that triggers the goal, the ID is recorded in the user's session data. Each time the user completes the action, a new session is created, and a new goal completion is recorded with a unique ID.
Therefore, if the user completes the newsletter sign-up three times in three separate sessions, analytics will count three goal conversions, each with a unique ID.
It's worth noting that while this behavior is typical for most analytics platforms, it's possible to configure goals in different ways, so the behavior may vary depending on the specific implementation. Therefore, it's important to consult the analytics documentation or work with a knowledgeable analyst to confirm the expected behavior.
Therefore, the correct answer is three goal conversions will be counted in analytics.
To know more about Analytics, refer here:
https://brainly.com/question/30101345#
#SPJ11
If a hospital patient is given 100 milligrams of medicine, which leaves the bloodstream at 14% per hour, how many milligrams of medicine will remain in the system after 10 hours? Use the function A(t) = Iert. 24. 66 mg 86. 94 mg 90. 48 mg 405. 52 mg.
Answers
what is the rule for the numbers 1; 2; 4; 7; 11; 16
Answers
Answer:
n²+n+2
Step-by-step explanation:
1)Find the difference between the numbers.
Difference= +1,+2,+3,+4,+5
2)Find the second difference between the first differences.
Difference= +1 each time
2 differences=Quadratic sequence
3)Quadratic Sequence Equation= an²+bn+c
Find the values of a, b and c by using these terms:
a (bottom)
2a+b (middle)
a+b+c (top)
4)Substitute values into equation
a=1 b=1 c=2
n²+n+2
A frame designer is making a triangular frame. She has two sides of length 18 inches and 27 inches. What are the possible lengths for the third side?.
Answers
The possible lengths( in whole elevation) for the third side is
9 elevation< x< 45 elevation, i.e in between 9 and 45 elevation.
For the below question, we've a rule from the properties of triangle, the sum of the length of any two sides of the triangle must be lesser than the length of the third side.
Hence
She has two sides of length 18 elevation and 27 elevation
Let the third side = x
Hence
a) 18 27> x
45> x
b) 18 x> 27
x> 27- 18
x> 9
thus, the possible lengths( in whole elevation) for the third side is
elevation< x< 45 elevation
Learn more about properties of triangle here :
https://brainly.com/question/30166682
#SPJ4
20 people each bring 3 food dishes to an awards banquet. The dishes are shared equally by 180 people at the banquet. How many people share 1 food dish?
Answers
Answer:
3 people
Step-by-step explanation:
If 20 people each bring 3 food dishes to an awards banquet, the total amount of food brought will be (20 * 3) = 60foods
If the dishes are shared equally by 180 people at the banquet, then;
60foods = 180 people
1 food = x
Cross multiply
60* x = 180
60x = 180
x =180/60
x = 3
Hence 3 people will share 1 dish
If point A is located at (-12, 6) on a coordinate plane, and point B is located at (4, 6), what is the distance between the two points? *
Answers
Answer:
the answer is (7,5) (5,5)
What is the area of this figure?
Answers
Answer:
30 yd²
Step-by-step explanation:
rectangle area = 6 x 4 = 24
triangle area = 1/2 x 3 x 4 = 6
24 + 6 = 30 yd²
also could be figured this way:
area = 1/2(b1 + b2)(h)
1/2(6 + 9)(4) = 30 yd²
now there are ten nucleotides floating around: a,a,a,a,t,c,c,c,g,g . at random, one is picked to go in the first position, and then another is picked to go in the second position, and then another is picked to go into the third position. what are the chances that the resulting sequence is cat ?
Answers
There will be 0.84% chances that resulting sequence in cat to occur.
Total number of nucleotides floating around = 10
Let the event of picking cat be "A"
Number of cat in group = 3
Probability of picking cat is picked at first position = P(A1) = 3 / 10
= 0.3
Probability of picking cat is picked at second position = P(A2) = 2 / 9
= 0.223
Probability of picking cat is picked at third position = P(A3) = 1 / 8
= 0.125
so , probability of picking cat in first position and again in second position and again in third position will be = P(A1) . P(A2) . P(A3)
= 0.3 × 0.223 × 0.125
= 0.0084 required probability
There will be 0.84% chances that resulting sequence in cat
To know more about Probability here
https://brainly.com/question/20370314
#SPJ4
Problem 3 A baseball is projected upward from the top of a 448-ft-tall building with an initial velocity of 48 feet per second. The hieght, h, of the baseball above the ground at any time, t, in seconds, is given by the equation h(t) = –16t2 + 48t + 448. a) Find the height after 2 seconds b) Find the height after 4 seconds c) Find the time it takes for the baseball to strike the ground.
Answers
a) The height of a baseball after 2 seconds = 480 ft
b) The height of a baseball after 4 seconds = 384 ft
c) The time taken by the baseball to strike the ground = 7 seconds
Here, the height , h, of the baseball above the ground at any time, t, in seconds, is given by the equation,
h(t) = -16t² + 48t + 448.
a) the height after 2 seconds
substitute t = 2 in above equation,
h(2) = -16(2)² + 48(2) + 448
h(2) = 480 ft
b) in order to the height after 4 seconds:
substitute t = 24 in above equation,
h(4) = -16(4)² + 48(4) + 448
h(4) = 384 ft
c) To find the time taken by the baseball to strike the ground, substitute h(t) = 0
So, -16t² + 48t + 448 = 0
We solve above quadratic equation.
(t + 4)(t - 7) = 0
t = -4 OR t = 7
t = -4 is not possible
Therefore, the time to strike the ground would be 7 seconds
Learn more about equation here:
https://brainly.com/question/649785
#SPJ4
The complete question is:
A baseball is projected upward from the top of a 448-ft-tall building with an initial velocity of 48 feet per second. The hieght, h, of the baseball above the ground at any time, t, in seconds, is given by the equation h(t) = -16t² + 48t + 448. a) Find the height after 2 seconds b) Find the height after 4 seconds c) Find the time it takes for the baseball to strike the ground.
A spinner is divided into five sections numbered 1 through 5. Juana records the number the spinner lands on for each of 50 spins. She computes and graphs the relative frequencies for each number. Which of the following describes the probability distribution? A bar graph entitled Relative Frequency of Numbers Spun has x on the x-axis and Probability on the y-axis. The probability of 1 is 0. 5; 2 is 0. 12; 3 is 0. 12; 4 is 0. 12; 5 is 0. 12. Constant symmetric negatively skewed positively skewed.
Answers
The distribution of the probability of the given data is a Positive Skewed Probability distribution.
Given to us
A spinner is divided into five sections numbered 1 through 5. Juana records the number the spinner lands on for each of the 50 spins. She computes and graphs the relative frequencies for each number
What is a Probability distribution?It is a type of distribution in which the distribution has a constant probability, therefore, the probability of each event occurring is equal.
What is symmetric distribution?Symmetric distribution is the type of distribution where the left side of the distribution and the right side of the distribution are mirror images to each other, therefore, they are symmetrical.
What is Negatively Skewed Probability distribution?Negatively Skewed Probability distribution is a type of distribution of the probability where the right side(tail) of the distribution is more concentrated than the right side.
What is Positive Skewed Probability distribution?Positive Skewed Probability distribution is a type of distribution of the probability where the left side of the distribution is more concentrated than the right side.
Which of the following describes the probability distribution of the given data?Since in the given graph probability of every number occurring from 2 to 5 is equal to 0.12(12%), while the probability of number 1 occurring is 0.5(50%).
Thus, the distribution of the probability is towards the right end of the graph.
Hence, the distribution of the probability of the given data is a Positive Skewed Probability distribution.
Learn more about Probability Distribution:
https://brainly.com/question/795909
Answer:b
b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b
Step-by-step explanation:
PLEASE HELP
Select the correct answer.
A six-sided fair die and an eight-sided fair die are rolled together. What is the probability of getting numbers whose sum is a multiple of 3?
A. 1/18
B. 1/4
C. 1/3
D. 4/9
E. 2/3
Answers
Answer:
C. 1/3
Step-by-step explanation:
there are 48 possible outcomes, and out of those 16 of them have a sum of 3.
16/48
simplify
1/3
If Wendy arranged 16 rows, she arranged a total of
books.
Answers
HELP PLEASE WILL MARK RIGHT ANSWER BRAINLIEST
Answers
Answer:
The last answer. The slope 1.8 is positive so it will slope upward from left to right. Since 1.8 is less than 2.5 the slope of the line will be less steep.
Which equation has a slope of-3 and passes through the point (-5,4)
Answers
1. Find the GCF for 14ab- 42a
Answers
Answer:
14(ab- 3a)
Step-by-step explanation:
14 times ab is 14ab and 14 times 3a is 42a
Yesterday was the first playoff game for the Culver City Cyclones baseball team. The
stadium's manager predicted attendance at the game would be 5,600 fans. Unfortunately, the
threat of a storm caused many to stay home, and his prediction wound up being 28% higher
than the actual attendance. What was the actual attendance at the baseball game?
Answers
The actual attendance at the baseball game is 7168 fans
What is meant by percentage?A figure or ratio that is stated as a fraction of 100 is referred to as a percentage in mathematics. The abbreviations "pct," "pct," and occasionally "pc" are also used to represent percentage in addition to the symbol "%". A% has neither a recognized unit of measurement nor any dimensions. Take this as an example: There will be 50 men in the class if there are 100 students total and there are 50 men in the class. There are 250 male students overall, or 250 out of 500.
Given,
The actual attendance at the game=5600 fans
Also given that, the threat of a storm caused many to stay home, and his prediction wound up being 28% higher than the actual attendance.
Then,
28% of 5600
=5600×(28/100)
=1568
Total=5600+1568
Total=7168
Therefore, the actual attendance at the baseball game is 7168 fans.
To know more about percentage, visit:
https://brainly.com/question/16797504
#SPJ1
Answer:
4,375 fans
Step-by-step explanation:
percent error= amount of error/correct amount
1.28x=5600
1.28x/1.28= 5000/1.28
x=4375
Define g(4) for the given function so that it is continuous at x = 4, 2x - 32 9(x) 2x - 8 Define g(4) as (Simplify your answer)
Answers
To ensures the function is continuous at x = 4, g(4) is equal to 136,
To define g(4) such that the function is continuous at x = 4, we need to find the value of g(4) that makes the function continuous at that point.
The given function is defined as: f(x) = 2x - 32, for x < 4 , f(x) = 9x^2 - 8, for x ≥ 4. To make the function continuous at x = 4, we set g(4) equal to the value of the function at that point. g(4) = f(4)
Since 4 is equal to or greater than 4, we use the second part of the function:
g(4) = 9(4)^2 - 8
g(4) = 9(16) - 8
g(4) = 144 - 8
g(4) = 136
Therefore, g(4) is equal to 136, which ensures the function is continuous at x = 4.
To know more about functions, refer here :
https://brainly.com/question/30721594#
#SPJ11
Dominique owns a flower shop she sold 36,000 flowers last year round the number of flowers she sold to the nearest 10,000 and the nearest 10,000
Answers
\((800)×(3×10 −1 )= ?\)
Answers
The answer is 23200.
BODMAS is an acronym that stands for Bracket, Order, Division, Multiplication, Addition, Subtraction. In certain regions, a synonym for BODMAS is used: PEMDAS (parentheses, exponents, multiplication, division, addition, and subtraction). It describes the order of operations performed when solving an equation. According to the BODMAS rule, if an expression contains parentheses ((), {}, []), first break or simplify the parentheses, then "order" (meaning exponentiation, root, etc.), division, multiplication , addition and subtraction from the left.By using BODMAS , we will first simplify the bracket term.
The bracket term is \((3\times10-1)\)
After , bracket we will operate the multiplication term which is given by
\(30-1\)
Now , we will use the subtraction operation .
\(29\)
On solving the bracket , we get 29 .
Now we will multiply 800 by 29 ,we get
\(800\times29=23200\)
Learn more about BODMAS here :
#SPJ9
The complete question is mention below :
Solve \(800(3\times10-1)\)
Name each vector, then write the vector in component form.
Answers
if vector B has components b₁ = 0, b₂ = 4, and b₃ = -2, we can write its component form as B = [0, 4, -2].
ToTo name a vector, we usually use a lowercase letter with an arrow above it or a boldface lowercase letter. However, since text-based communication does not support special characters or formatting, I will simply name each vector using a letter.
Let's consider two vectors:
Vector A:
Component form: A = [a₁, a₂, a₃]
Vector B:
Component form: B = [b₁, b₂, b₃]
In the component form, each vector is represented by an ordered list of its components. The subscript indicates the position of each component in the vector.
For example, if vector A has componentcomponentss a₁ = 2, a₂ = -1, and a₃ = 3, we can write its component form as A = [2, -1, 3].
Similarly, if vector B has components b₁ = 0, b₂ = 4, and b₃ = -2, we can write its component form as B = [0, 4, -2].
Please note that the specific values for the components of the vectors will determine the actual component form.
To learn more about vector click here:brainly.com/question/30958460
#SPJ11
Find the value of x.
Answers
Answer:
x = 46
Step-by-step explanation:
These 2 angles would add up to 180 degrees.
So 41 + (3x+1) = 180
Let's solve for x.
41 + 3x + 1 = 180
Combine like terms.
42 + 3x = 180
subtract 42 from both sides
3x = 180-42
3x = 138
divide both sides by 3
x = 46
Which equation represents a line which is parallel to the line 3x + 8y = - 40 ?
Answers
It’s just an attempt
what is 372777177771136661331x132747717743716662141284957x0=
Answers
Answer:
0
Step-by-step explanation:
Anything times zero, is zero.
Solve for b pls. Due by 2:45.....
Answers
Answer:
b = 12
Step-by-step explanation:
Using Pythagoras' identity in the right triangle
b² + 5² = 13²
b² + 25 = 169 ( subtract 25 from both sides )
b² = 144 ( take the square root of both sides )
b = \(\sqrt{144}\) = 12
Find the square of 111 base two give ur answer in base two
Answers
Answer: I'm not 100% sure what your asking but I believe the answer is 12321.
Step-by-step explanation:
what is the major assumption about the distribution of returns that we have to make to get to this objective function?
Answers
To arrive at the objective function, we must make the main assumption that the returns are normally distributed
What main assumption should we make?The objective function is a mathematical expression used to represent the goal of a problem in terms of one or more variables. In finance, the objective function is used to represent the goal of maximizing returns.
The distribution of returns is an important factor to consider when creating an objective function for an investment portfolio. Returns are the gains or losses an investor makes on his investments. The distribution of returns refers to the way in which these returns are distributed in the portfolio.
To arrive at the objective function, we must make the main assumption that the returns are normally distributed. A normal distribution is a bell-shaped curve that represents the distribution of data in a population. In finance, normal distributions are used to model portfolio returns because they ensure a good approximation of how returns are distributed across a wide range of investments.
See more information about objective function in: https://brainly.com/question/26100401
#SPJ11
Solve the following proportions
Answers
Cross multiply
40.3m = 6.5 x 12.4
40.3m = 80.6
Divide through by 40.3
m = 80.6/40.3
m= 2
m= 2
Answer:
\(6.5 \times 12.4 = 40.3 \: m \\ \frac{80.6}{40.3} = 2\)