Which solids can have vertical cross sections that are circles? Check all that apply
-cones
-cylinders
-spheres
cones
cylinders
spheres
Step-by-step explanation:
The question was worded incorrectly and instead of giving the options it gave you the answers
A golf ball is selected at random from a golf bag. If the golf bag contains 5 type A balls, 8 type B balls, and 3 type C balls, find the probability that the golf ball is not a type A ball.
Final answer:
The probability that a randomly selected golf ball from the bag is not a type A ball is 11/16, as we calculate this by dividing the number of non-type A balls (11) by the total number of balls (16).
Explanation:
The probability that the golf ball selected at random is not a type A ball can be found by first determining the total number of balls in the golf bag and then subtracting the number of type A balls to obtain the number of non-type A balls. The total number of balls is 5 type A balls + 8 type B balls + 3 type C balls = 16 balls. The number of non-type A balls is 8 type B balls + 3 type C balls = 11 balls.
To find the probability that the selected ball is not a type A ball, we divide the number of non-type A balls by the total number of balls, which gives us a probability of 11/16.
Final answer:
The probability that a randomly selected golf ball from the bag is not a type A ball is 0.6875 or 68.75%.
Explanation:
To find the probability that the golf ball selected at random is not a type A ball, we need to determine the total number of non-type A balls in the golf bag and divide it by the total number of balls in the bag. The golf bag contains 5 type A balls, 8 type B balls, and 3 type C balls, so the total number of balls in the bag is 5 + 8 + 3 = 16. There are 8 + 3 = 11 non-type A balls (type B and type C).
The probability of selecting a non-type A ball is then given by the number of non-type A balls divided by the total number of balls:
Probability = Number of non-type A balls / Total number of balls
We calculate it as:
Probability = 11 / 16 = 0.6875
Thus, the probability that the selected golf ball is not a type A ball is 0.6875 or 68.75%.
A grocery store has an average sales of $8000 per day. The store introduced several advertising campaigns in order to increase sales. To determine whether or not the advertising campaigns have been effective in increasing sales, a sample of 64 days of sales was selected. It was found that the average was $8300 per day. From past information, it is known that the standard deviation of the population is $1200. The correct null hypothesis for this problem is
Answer:
We need to conduct a hypothesis in order to check if the true mean for sales is significantly higher than 8000, the system of hypothesis would be:
Null hypothesis:[tex]\mu \leq 8000[/tex]
Alternative hypothesis:[tex]\mu > 8000[/tex]
[tex]z=\frac{8300-8000}{\frac{1200}{\sqrt{64}}}=2[/tex]
[tex]p_v =P(z>2)=0.0228[/tex]
Step-by-step explanation:
Data given
[tex]\bar X=8300[/tex] represent the sample mean for the sales
[tex]\sigma=1200[/tex] represent the population standard deviation
[tex]n=64[/tex] sample size
[tex]\mu_o =8000[/tex] represent the value that we want to test
z would represent the statistic (variable of interest)
System of hypothesis
We need to conduct a hypothesis in order to check if the true mean for sales is significantly higher than 8000, the system of hypothesis would be:
Null hypothesis:[tex]\mu \leq 8000[/tex]
Alternative hypothesis:[tex]\mu > 8000[/tex]
The statistic to check this hypothesis is given by:
[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
Calculate the statistic
[tex]t=\frac{8300-8000}{\frac{1200}{\sqrt{64}}}=2[/tex]
P-value
Since is a one right tailed test the p value would be:
[tex]p_v =P(z>2)=0.0228[/tex]
Based on these tables, what can you determine about the students in this class? Check all that apply.
There are 35 students in the class.
11 of the students are boys who have summer birthdays.
19 of the students are boys.
There is not enough information shown to determine how many girls have summer birthdays.
Answer:
Step-by-step explanation:
It’s 1 3,and 4
Answer:
1,3,4
Step-by-step explanation:
How do I do a triangles heirarchy
Answer:
measure each side of the triangle and make sure it is right and then graph is out.
Step-by-step explanation:
what is 81,007-26,318?
Answer:
54689
Step-by-step explanation:
A random sample of 28 plastic items is obtained, and their breaking strengths are measured. The sample mean is 7.142 and the sample standard deviation is 0.672. Conduct a hypothesis test to assess whether there is evidence that the average breaking strength is not 7.000.
Answer:
The test statistic t = 1.126 < 1.703 of '27' degrees of freedom at 0.05 level of significance.
null hypothesis(H₀ ) is accepted
There is evidence that the average breaking strength is 7.000.
Step-by-step explanation:
Step 1:-
Given random sample size (n) =28 <30
small sample size n= 28
The sample mean (x⁻) = 7.142
sample standard deviation (S) =0.672
Step 2:-
Null hypothesis :- there is evidence that the average breaking strength is 7.000.
H₀ : μ =7
Alternative hypothesis:-there is evidence that the average breaking strength is 7.000.
H₁ : μ ≠7
The test statistic [tex]t = \frac{x^{-} -mean}{\frac{S}{\sqrt{n} } }[/tex]
Substitute all values and simplification ,
[tex]t = \frac{7.142 -7}{\frac{0.672}{\sqrt{28} } } = \frac{0.142 }{0.1269}[/tex]
t = 1.126
Calculated value is t = 1.126
The degrees of freedom γ = n-1 = 28-1 =27
The tabulated value t= 1.703 at degrees of freedom at 0.05 level of significance.
since calculated t < tabulated value 't' value of 27 degrees of freedom at 0.05 level of significance.
null hypothesis(H₀ ) is accepted
There is evidence that the average breaking strength is 7.000.
The radius of a circle is 4 yards. What is its circumference?
Answer:
See answer below
Step-by-step explanation:
Hi there,
To get started, recall the circumference formula.
[tex]C = \pi r^{2}[/tex] where π is the irrational number 3.14159... and r is the radius of the circle. Circumference is like perimeter but for a circle; it is the distance around the boundary.
[tex]C = \pi (4 \ yd)^{2}= 16\pi \ yd^{2} = 50.27 \ yd^{2}[/tex] (approximately)
thanks,
Answer:
C≈25.13yd
Step-by-step explanation:
The circumference of a circle can be found by multiplying pi ( π = 3.14 ) by the diameter of the circle. If a circle has a diameter of 4, its circumference is 3.14*4=12.56. If you know the radius, the diameter is twice as large.
Hope that was helpful.Thank you!!!
In a certain region, 15% of people over the age of 50 didn’t graduate from high school. We would like to know if this percentage is the same among the 25-50 year age group. What is the minimum number of 25-50 year old people who must be surveyed in order to estimate the proportion of non-grads to within 6% of the true parameter with 99% confidence?
Answer:
235 people
Step-by-step explanation:
Given:
P' = 15% = 0.15
1 - P' = 1 - 0.15 = 0.85
At 99% confidence leve, Z will be:
[tex] \alpha [/tex] = 1 - 99%
= 1 - 0.99 = 0.01
[tex] \alpha /2 = \frac{0.01}{2} = 0.005 [/tex]
[tex] Z\alpha/2 = 0.005 [/tex]
Z0.005 = 2.576
For the minimum number of 25-50 year old people who must be surveyed in order to estimate the proportion of non-grads to within 6%, we have:
Margin of error, E = 6% = 0.06
sample size = n = [tex] (\frac{Z\alpha /2}{E})^2 * P* (1 - P) [/tex]
[tex] = (\frac{2.576}{0.06}) ^2 * 0.15 * 0.85 [/tex]
= 235.02 ≈ 235
A number of 235 people between 25-30 years should be surveyed .
Answer:
n = 236
Step-by-step explanation:
Solution:-
- The proportion of people over the age of 50 who didn't graduate from high school are, p = 0.15 - ( 15 % )
- We are to evaluate the minimum sample size " n " from the age group of 25-50 year in order to estimate the proportion of non-grads within a standard error E = 6% of the true proportion p within 99% confidence.
- The minimum required sample size " n " for the standard error " E " for the original proportion p relation is given below:
[tex]n = \frac{(Z_\alpha_/_2)^2 * p* ( 1 - p )}{E^2}[/tex]
- The critical value of standard normal is a function of significance level ( α ), evaluated as follows:
significance level ( α ) = ( 1 - CI/100 )
= ( 1 - 99/100 )
= 0.01
- The Z-critical value is defined as such:
P ( Z < Z-critical ) = α / 2
P ( Z < Z-critical ) = 0.01 / 2 = 0.005
Z-critical = Z_α/2 = 2.58
- Therefore the required sample size " n " is computed as follows:
[tex]n = \frac{(2.58)^2 * 0.15* ( 1 - 0.15 )}{0.06^2}\\\\n = \frac{6.6564 * 0.1275}{0.0036}\\\\n = \frac{0.848691}{0.0036}\\\\n = 235.7475\\[/tex]
Answer: The minimum sample size would be next whole number integer, n = 236.
An Individual Retirement Account (IRA) has $17 comma 000in it, and the owner decides not to add any more money to the account other than interest earned at 4%compounded daily. How much will be in the account 30years from now when the owner reaches retirement age?
Answer: The owner reaches at Rs. 56438.28 after 30 years.
Step-by-step explanation:
Since we have given that
Sum = Rs. 17000
Rate of compounded daily = 4%
Number of years = 30 years
So, Using "compound interest formula" we get that :
[tex]A=P(1+\dfrac{r}{n})^{nt}\\\\A=17000(1+\dfrac{0.04}{365})^{365\times 30}\\\\A=17000(1.000109589)^{10950}\\\\A=56438.28[/tex]
Hence, The owner reaches at Rs. 56438.28 after 30 years.
Find the volume of the cone
Answer:
209.467mm³
Step-by-step explanation:
the explanation is in the picture
hope this helps<3
please like and Mark as brainliest
An experiment is carried out 400 times the possible outcomes are void fail and success if the frequency of void is 96 and the relative frequency is 0.24 then how much is the frequency of fail and success
the frequency of success is 244.
part b's answer is 240.
A relative frequency distribution shows the proportion of the total number of observations associated with each value or class of values and is related to a probability distribution, which is extensively used in statistics.
Relative frequency can be defined as the number of times an event occurs divided by the total number of events occurring in a given scenario. The relative frequency formula is given as:
Relative Frequency = Subgroup frequency/ Total frequency.
1. Frequency of success
=400 - 96 - 60
=244
relative frequency of fail
=60/400= 0.15
Relative Frequency of success
=1-0.15 - 0.24
=0.61
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Paulo works at the United Nations. He researched what percent of the world's population lives on each continent. He surveys a sample of employees at the United Nations about their continent of origin to see if the distribution in the sample agrees with the percentages he researched.
Which of these inference procedures is most appropriate?
Answer:
confidence interval using a two sample t test between percents
Step-by-step explanation:
confidence interval using a two sample t test between percents This can be used to compare percentages drawn from two independent samples in this case employees. It is used to compare two sub groups from a single sample example the population on the planet
Choose all that are correct. Choosing the brainliest.
Answer:
A, B and F
Step-by-step explanation:
Area of 2 triangles:
2(½ × 6 × 8)
48 in²
Area of 3 rectangles:
3(18 × 6)
324 in²
Total surface area:
324 + 48
372 in²
1 poir
Elizabeth's tablet has a combined total of 20 apps and movies. Let x
represent the number of apps and y represent the number of movies.
Which of the following could represent the number of apps and movies on
Elizabeth's tablet? Select all that apply.
The given options are:
(A)x+y = 20 (B)7 apps and 14 movies (C)x-y= 20 (D)y=-x+ 20 (E)8 apps and 12 movies (F)xy= 20Answer:
(A)x+y = 20 (D)y=-x+ 20 (E)8 apps and 12 moviesStep-by-step explanation:
If Elizabeth has a combined total of 20 apps and movies.
Where:
Number of apps=x
Number of Movies =y
Then:
Their total,
x+y=20 (Option A)If we subtract x from both sides
x+y-x=-x+20
y=-x+20 (Option D)In Option E
8 apps and 12 movies add up to 20. Therefore, this could also apply.
Braydon, a scuba diver, has a tank that holds 6 liters of air under a pressure of 220 pounds per square inch (psi).
Write the equation that relates pressure, P, to volume, V.
If the pressure increases to 330 psi, how much air is held in Braydon’s tank?
Answer: 4 litres of air is held in Braydon’s tank.
Step-by-step explanation:
The law relating pressure to volume is the Boyle's law. It states that the volume of a given mass of gas is inversely proportional to its pressure as long as temperature remains constant. It is expressed as
P1V1 = P2V2
Where
P1 and P2 are the initial and final pressures of the gas.
V1 and V2 are the initial and final volumes of the gas.
From the information given,
V1 = 6 litres
P = 220 psi
P2 = 330 psi
Therefore,
6 × 220 = 330V2
V2 = 1320/330 = 4 litres
Answer:
V=1320/p
the tank holds 4 liters
Step-by-step explanation:
Find the arc length of the following curve on the given interval. x equals 8 t minus 7 comma y equals 15 t minus 6x=8t−7, y=15t−6, 0 less than or equals t less than or equals 40≤t≤4 The length of the curve is nothing. (Type an integer or a fraction.)
Answer:
612
Step-by-step explanation:
Both x and y are linear functions of t, so for each increment of t, the x- and y-coordinates will increment by 8 and 15, respectively. The length of a line segment joining points 8 units in the horizontal direction and 15 units in the vertical direction is given by the Pythagorean theorem as ...
d = √(8² +15²) = 17
From t=4 to t=40, there are 36 increments in t, so the length of the line segment defined by the given functions is ...
36×17 = 612 . . . units
part of a $3,600 bonus was invested at 9% annual simple interest. The rest was invested at %8 annual simple interest. The total interest at the end of one year was $312. How much was invested in the %9 account?
Answer:
$2400
Step-by-step explanation:
Let x represent the amount invested at 9%. (3600-x) will be the amount invested at 8%. The total interest earned is then ...
312 = 0.09x +0.08(3600 -x)
24 = .01x . . . . subtract 288, simplify
2400 = x . . . . divide by .01
$2400 was invested in the 9% account.
The population in the city of Millstone was approximately 2 million in 2010 and 2.2 million in 2015. What is the percent increase from 2010 to 2015
Answer:
10%
Step-by-step explanation:
Percentage increase for any change is calculated by formula
{(Final value - initial value)/initial value} * 100
Given
population in 2010 = 2 million ----------->initial value
population in 2015 = 2.2 million ----------->Final value
[tex]Percentage \ \ increase = {(2.2 - 2)/2} *100= (0.2/2)*100 = 10%[/tex]
Answer:
10%
Step-by-step explanation:the population in 2010 - 2015 is grown though 10%
as a percent incease the number willl incease in increase will increase and increase until the number can increase to 2.2 million or 2 millions and also the population would probably decrease
I NEED HELP DUE IN 2 MINS!!
Answer:Triangle A
Step-by-step explanation: It has one right angle.
Answer:
THE ANSWERS A
Step-by-step explanation:
OMG GOOD LUCK!!! BECAUSE IT HAS THE AREA LAYED OUT HAVE A BLESSED DAY!!!
3x+5=3 solve this equation
Answer:
-3/2
Step-by-step explanation:
Answer: x = -2/3 = -0.667
Suppose that $2n$ tennis players compete in a round-robin tournament. Every player has exactly one match with every other player during $2n-1$ consecutive days. Every match has a winner and a loser. Show that it is possible to select a winning player each day without selecting the same player twice. \\ \\ \textit{Hint: Remember Hall's Theorem}
Answer:
Step-by-step explanation:
given that Suppose that $2n$ tennis players compete in a round-robin tournament. Every player has exactly one match with every other player during $2n-1$ consecutive days.
this is going to be proved by contradiction
Let there be a winning player each day where same players wins twice, let n = 3there are 6 tennis players and match occurs for 5daysfrom hall's theorem, let set n days where less than n players wining a day let on player be loser which loses every single day in n days so, players loose to n different players in n daysif he looses to n players then , n players are winnerbut, we stated less than n players are winners in n days which is contradiction.so,we can choose a winning players each day without selecting the same players twice.Directions for questions 4 & 5: We selected a random sample of 100 StatCrunchU students, 67 females and 33 males, and analyzed their responses to the question, "What is the total amount (in dollars) of credit card debt you have accrued to date?" With more than 30 in each random and independent sample, conditions are met for modeling the distribution of differences in sample means using a T-model. Therefore, we will proceed with finding a confidence interval to estimate the gender difference in credit card debt for StatCrunchU students. Summary statistics for CC Debt: Group by: Gender Gender Mean Std. dev. n Female 2577.75 1916.29 67 Male 3809.42 2379.47 33 Use StatCrunch to find the 95% confidence interval estimating the difference µ1 – µ2, where µ1 is the mean credit card debt for all female StatCrunchU students and µ2 is the mean credit card debt for all male StatCrunchU students. (directions) Since the numbers are dollars, round to two decimal places when you enter your answer. Flag this Question Question 42 pts The lower limit on the confidence interval is
The lower limit of the 95% confidence interval for the difference in mean credit card debt between female and male students can be calculated by formula using sample means, standard deviations, sample sizes, and accounting for the t-value associated with 95% confidence.
Explanation:To calculate the 95% confidence interval for the difference between the mean credit card debt of female and male StatCrunchU students, we use the given information: µ1 (mean credit card debt of females) = 2577.75, µ2 (mean credit card debt of males) = 3809.42, std. dev. of females = 1916.29, std. dev. of males = 2379.47, number of females = 67, number of males = 33.
To calculate the confidence interval, we will use the t-model formula for confidence intervals for difference in means, which is:
(µ1-µ2) ± t*(sqrt([std. dev.1/sqrt(n1)] + [std. dev.2/sqrt(n2)]))
After plugging in the objective values, we would get the confidence range. The lower limit will be (µ1-µ2) - t*(sqrt([std. dev.1/sqrt(n1)] + [std. dev.2/sqrt(n2)])).
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Lines q and r are parallel.
Parallel lines q and r are cut by transversals s and t. The angles formed by the intersection of lines q, s, and t, clockwise from top left, are blank, 53 degrees, blank, 57 degrees, blank, blank; formed by s and r are blank, 5 x degrees, blank, blank.
What is the value of x?
14
22
53
70
Answer:
x=22
Step-by-step explanation:
The value of x is 14.
What is Coordinate System?Arrangement of reference lines or curves used to identify the location of points in space.
Given that Parallel lines q and r are cut by transversals s and t.
The angles formed by the intersection of lines q, s, and t, clockwise from top left, are blank, 53 degrees, blank, 57 degrees,
We need to find the value of x.
The unknown angle between 53 and 57 be u
53+57+u=180
110+u=180
u=70
Now this angle is corresponding to 5x
5x=70
Divide both sides by 5
x=14
Hence, the value of x is 14.
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You have inherited land that was purchased for $40,000 in 1990. The value of the land
increased by
approximately 5% per year. Which relationship would represent the value of the land in the
year 2020
Answer:
[tex]A(30)=40000(1.05)^{30}[/tex]
Step-by-step explanation:
Given that the land was purchased for $40,000 in 1990, the initial amount/principal =$40,000
Since its value increases by approximately 5% per year, we can model this growth using the compound interest formula:
[tex]A=P(1+r)^n[/tex]
P=$40,000, r=5%=0.05, n=2020-1990=30 Years
Therefore, we have the value of the land in 30 years time to be:
[tex]A=40000(1+0.05)^{30}\\A(30)=40000(1.05)^{30}[/tex]
Since the options are not available, the relationship which represents the value of the land in the year 2020 is:
[tex]A(30)=40000(1.05)^{30}[/tex]
For which survey is a sample not necessary?
What percentage of Colorado residents support planting more trees in the community?
Which electronic gadget will be the most popular among middle school students this year?
Do your classmates prefer warm or cool places to travel for vacation?
How many car accidents involve air bag malfunctions?
Answer:
c
Step-by-step explanation:
Answer:
Do your classmates prefer warm or cool places to travel for vacation?
Step-by-step explanation:
A recent study reported that 18- to 24-year-olds average 192 restaurant visits per year. Assume that the standard deviation for number of visits per year for this age group is 56.5. To validate these findings, a random sample of forty 18- to 24-year-olds was selected and found to average 212 restaurant visits per year. Which of the following statements is correct
A.)The interval that contains 95% of the sample means is 170.3 and 213.7 visits. Because the sample mean is between these two values, we have support for the results of the May 2011 study.
B.)The interval that contains 95% of the sample means is 170.3 and 213.7 visits. Because the sample mean is between these two values, we do not have support for the results of the May 2011 study.
C.)The interval that contains 95% of the sample means is 174.5 and 209.5 visits. Because the sample mean is not between these two values, we have support for the results of the May 2011 study.
D.)The interval that contains 95% of the sample means is 174.5 and 209.5 visits. Because the sample mean is not between these two values, we do not have support for the results of the May 2011 study.
Answer:
Option C) is the correct answer.
Step-by-step explanation:
We are given the following in the question:
Mean = 192
Sample mean, [tex]\bar{x}[/tex] = 212
Sample size, n = 40
Alpha, α = 0.05
Population standard deviation, σ = 56.5
95% Confidence interval:
[tex]\mu \pm z_{critical}\dfrac{\sigma}{\sqrt{n}}[/tex]
Putting the values, we get,
[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]
[tex]192 \pm 1.96(\dfrac{56.5}{\sqrt{40}} ) = 192 \pm 17.5 = (174.5,209.5)[/tex]
Thus, the correction answer is
Option C)
"The interval that contains 95% of the sample means is 174.5 and 209.5 visits. Because the sample mean is not between these two values, we have support for the results of the May 2011 study."
Consider rolling two number cubes, each of which has its faces numbered
from 1 to 6. The cubes will be rolled and the sum of the numbers landing
face up will be recorded. Let the event E represent the event of rolling a sum
of 5. How many outcomes are in the collection for event E?
Answer: 4 outcomes
Step-by-step explanation:
For two number cubes, the total possible outcomes are:
6 events for the first and 6 events for the second, then the total number of combinations is 6*6 = 36
If the dice are different, the possible outcomes are:
Dice 1 = 3, Dice 2 = 2
Dice 1 = 2, Dice 2 = 3
Dice 1 = 4, Dice 2 = 1
Dice 1 = 1, Dice 2 = 4
Then we have 4 outcomes in the collection for event E.
Answer:
4
Step-by-step explanation:
I did it on college board
Meta-analysis involves:
a. averaging all the test statistics from every possible study on a given topic.
b. finding all studies published on a topic, calculating the effect size for each of those studies, and averaging the effect sizes together to find the average size of the effect across all studies.
c. finding all studies published on a topic, contacting the authors of the studies to request their original data, and then analyzing all the obtained data in one large analysis of variance.
d. attempting to recreate the experimental conditions of every published study on a given topic.
Answer: b. finding all studies published on a topic, calculating the effect size for each of those studies, and averaging the effect sizes together to find the average size of the effect across all studies.
Step-by-step explanation:
For problems 13-17 find a particular solution of the nonhomogeneous equation, given that the functions y1(x) and y2(x) are linearly independent solutions of the corresponding homogeneous equation. x^2y''+xy'-4y=x(x+x^3)
Answer:
y_g(x) = C1*x^2 + C2*x^-2 + x^4 / 12
Step-by-step explanation:
Given:-
- The following second order ODE :
x^2y''+xy'-4y=x*(x+x^3)
Find:-
Find a particular solution of the nonhomogeneous equation
Solution:-
- First note that the ODE given is a Cauchy Euler ODE. The order of derivative of independent and dependent variables are similar. The general form of Cauchy Euler ODE is:
a*x^n y^(n) + b*x^n-1 y^(n-1) + c*x^n-2 y^(n-2) + ... + d*y = f(x)
- We will use the following Auxiliary Equation to find the complementary solutions - Solving Homogeneous part of ODE.
am*(m-1) + bm + c = 0
Where, a,b,c are constants such that:
x^2y'' + xy' - 4y = 0
a = 1 , b = 1 , c = -4
- Solve the Auxiliary equation for (m) as follows:
m*(m-1) + m - 4 = 0
m^2 - 4 = 0
m = +/- 2 ...... ( Real and distinct roots )
- The complementary solutions to the Real and distinct roots from Auxiliary Equation is:
yc(x) = y1(x) + y2(x)
yc(x) = C1*x^2 + C2*x^-2 .... ( Complementary Solution ).
- Now for the non-homogeneous part of ODE. The function f(x) is defined as:
f(x) = x*( x + x^3 ) = x^2 + x^4
- We see that (x^2) term is common to both f(x) and complementary solution yc(x). So when we develop a particular solution, we have to make sure that the solution is independent from complementary solution. If not we multiply the particular solution with (x^n). Where n is the smallest possible integer for which the solution is independent. So in our case ( Using undetermined Coefficient method ) :
y_p (x) = A*x^4 + B*x^3 + C*x^2 + D*x + E
- To make the solution independent we multiply y_p by (x^3) where n = 3.
y_p (x) = A*x^7 + B*x^6 + C*x^5 + D*x^4 + E*x^3
- Take first and second derivatives of the y_p(x) as follows:
y'_p(x) = 7A*x^6 + 6B*x^5 + 5C*x^4 + 4D*x^3 + 3E*x^2
y''_p(x) = 42Ax^5 + 30Bx^4 + 20Cx^3 + 12Dx^2 + 6Ex
- Substitute y_p(x) , y'_p(x) and y''_p(x) into the ODE given:
42Ax^7 + 30Bx^6 + 20Cx^5 + 12Dx^4 + 6Ex^3
+ 7Ax^7 + 6B*x^6 + 5C*x^5 + 4D*x^4 + 3E*x^3
- ( 4Ax^7 + 4B*x^6 + 4C*x^5 + 4D*x^4 + 4E*x^3 )
--------------------------------------------------------------------------------
45Ax^7 + 32Bx^6 + 21Cx^5 + 12Dx^4 + 5Ex^3
---------------------------------------------------------------------------------
45Ax^7 + 32Bx^6 + 21Cx^5 + 12Dx^4 + 5Ex^3 = x^2 + x^4
- Compare the coefficients:
A = B = C = E = 0
D = 1 / 12.
The particular solution is:
y_p(x) = x^4 / 12
- The general solution is as follows:
y_g(x) = yc(x) + y_p(x)
y_g(x) = C1*x^2 + C2*x^-2 + x^4 / 12
Answer:
The particular solution to the differential equation
x²y'' + xy' - 4y = x(x + x³)
is
y_p = (1/12)x^4 - x²/2 - x/3
Step-by-step explanation:
Given the differential equation:
x²y'' + xy' - 4y = x(x + x³)...............(1)
First, we solve the homogeneous part of (1)
x²y'' + xy' - 4y = 0...........................(2)
Let x = e^z
=>z = lnx
Let D = d/dz
dz/dx = (1/x)
dy/dx = (dy/dz).(dz/dx)
= (1/x)(dy/dz)
dy/dz = xdy/dx = xy' = Dy
d²y/dx² = (-1/x²)(dy/dz) + (1/x)(d²y/dx²)(dz/dx)
= (1/x²)(d²y/dx² - dy/dz) = (1/x²)(D² - D)y
Using these, (2) becomes
(D² - D)y + Dy - 4y = 0
(D² - 4)y = 0
The auxiliary equation is
m² - 4 = 0
(m - 2)(m + 2) = 0
m1 = 2, m2 = -2
The complementary function is
y = C1e^(2z) + C2e^(-2z)
But z = lnx
y_c = C1x² + C2/x² ...........................(3)
Now we solve (1) using the method of undetermined coefficients.
The nonhomogeneous part is
x(x + x³) = x² + x^4
So, we assume a particular solution of the form
y_p = Ax^4 + Bx³ + Cx² + Dx + E
y'_p = 4Ax³ + 3Bx² + 2Cx + D
y''_p = 12Ax² + 6Bx + 2C
Using these in (1)
x²y''_p + xy'_p - 4y_p = x²(12Ax² + 6Bx + 2C) + x(4Ax³ + 3Bx² + 2Cx + D) - 4(Ax^4 + Bx³ + Cx² + Dx + E)
= x² + x^4
12Ax^4 + 6Bx³ + 2Cx + 4Ax^4 + 3Bx³ + 2Cx² + Dx - 4Ax^4 - 4Bx³ - 4Cx² - 4Dx - 4E = x² + x^4
Comparing the coefficients of various powers of x, we have
12A + 4A - 4A = 1
12A = 1
=> A = 1/12
6B + 3B - 4B = 0
5B = 0
=> B = 0
2C - 4C = 1
-2C = 1
=> C = -1/2
2C + D - 4D = 0
2C - 3D = 0
2C = 3D
2(-1/2) = 3D
=> D = -1/3
-4E = 0
=> E = 0
(A, B, C, D, E) = (1/12, 0, -1/2, -1/3, 0)
y_p = Ax^4 + Bx³ + Cx² + Dx + E
= (1/12)x^4 - (1/2)x² - (1/3)x
The general solution is
y = y_c + y_p
= C1x² + C2/x² + (1/12)x^4 - x²/2 - x/3