Find a general solution of the following ODE using
Frobenius method.
x(x-1)y" + (3x-1)y' + y = 0
Answers
The general solution of the ODE x(x-1)y" + (3x-1)y' + y = 0 using the Frobenius method is asy = c₁x + c₂x² + x³(2c₁ - c₂)/3.
Consider a second-order linear differential equation in the form below:
xy'' + p(x)y' + q(x)y = 0
Here, p(x) and q(x) are power series about x = 0 i.e.,
p(x) = Σpₙxⁿ and q(x) = Σqₙxⁿ
So, y = Σaₙxⁿ is a power series about x = 0
Now, substitute y = Σaₙxⁿ into the given differential equation and simplify the equation and we get,
∑ₙ(xⁿ(x-1)aₙ⁺² - xⁿ(1-2n)aₙ + aₙ-₂(3n-a-1)aₙ) = 0
Here, the coefficient of each power of x must be zero individually.
For n = 0, we get the value of a₂ as a₂ = 0
Now, for n > 0 the indicial equation is,
x(x-1)n(n-1) + (3x-1)n - 2n = 0
⇒ n(n-1) + (3-2n)n - 2n = 0
⇒ n² - n = 0
⇒ n(n-1) = 0
So, n = 0, 1.
For n = 0, we have already calculated the value of a₂ as 0.
Now, for n = 1, we get the value of a₃ asa₃ = [2a₁ - a₀]/3and the solution of the given ODE is
y = a₀ + a₁x + (2a₁-a₀)x³/3
The general solution is given asy = c₁x + c₂x² + x³(2c₁ - c₂)/3
where c₁ and c₂ are arbitrary constants.
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Related Questions
Solve : 6 yards 9 feet 4 inches times 2
Answers
Answer:
18yrds n 8 inches
State if the two triangles are congruent. If they are, state how you know.
Answers
Answer: yes, aas
Step-by-step explanation:
They have the two angles, and then the side after the angles
water leaks from a tank at a rate of 2 8t liters per hour, where t is the number of hours after 7 {\tiny{am}}. how much water is lost between 9 and 11 {\tiny{am}}?
Answers
The water that was lost between 9 and 11 am is 52 liters by using integration.
What is meant by integration?An integral in mathematics assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that come from merging infinitesimal data. The process of determining integrals is known as integration. Along with differentiation, integration is a fundamental, essential operation of calculus and is used to answer issues in mathematics and physics involving the area of an arbitrary form, the length of a curve, and the volume of a solid, among others.
The integrals listed below are definite integrals, which can be defined as the signed area of the plane region circumscribed by the graph of a given function between two points on the real line.
Rate of leaking R'(t)=2+8t
Here t is the number of hours.
Total water lost from 9 and 11 am
=\(\int\limits^8_2\) (2+8t)dt
=(2t+(8t²/2))⁴₂
=(8+64)-(4+16)
=72-20
=52 liters.
The water that was lost between 9 and 11 am is 52 liters.
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X-32>32+9x Thx help ASAP
Answers
Answer:
Step-by-step explanation:
hello :
x-32>32+9x
means : x-9x>32+32
-8x >64
dvid by :8 -x >8 .......continu
the datafor each grade have the same interquartile range. which of the following best compares the twotest score distribution?
Answers
We are given the dot-plots of sixth-grade test scores and seventh-grade test scores.
Let us first find the median of the two test scores.
Recall that the median is the value that divides the distribution into two equal halves.
Sixth Grade Geograph Test Scores:
From the dot-plot, we see that 11 is the median test score since it divides the distribution into two equal halves.
Median = 11
Seventh Grade Geograph Test Scores:
From the dot-plot we see that 13 is the median test score since it divides the distribution into two equal halves.
Median = 13
Therefore, the median score of the seventh-grade class is 2 points greater than the median score of the sixth-grade class.
Now let us find the interquartile range which is given by
\(IQR=Upper\: quartile-Lower\: quartile\)Seventh Grade Geograph Test Scores:
The upper quartile is given by
\(Upper\: quartile=\frac{3}{4}(\operatorname{median})=\frac{3}{4}(13)=9.75=10th\text{ }\)At the 10th position, we have a test score of 13
The lower quartile is given by
\(Lower\: quartile=\frac{1}{4}(\operatorname{median})=\frac{1}{4}(13)=3.25=4th\)At the 3rd position, we have a test score of 11
So, the interquartile range is
\(IQR=Upper\: quartile-Lower\: quartile=13-11=2\)Sixth Grade Geograph Test Scores
The upper quartile is given by
\(Upper\: quartile=\frac{3}{4}(\operatorname{median})=\frac{3}{4}(11)=8.25=9th\text{ }\)At the 9th position, we have a test score of 10
The lower quartile is given by
\(Lower\: quartile=\frac{1}{4}(\operatorname{median})=\frac{1}{4}(11)=2.75=3rd\)At the 3rd position, we have a test score of 8
So, the interquartile range is
\(IQR=Upper\: quartile-Lower\: quartile=10-8=2\)So, the IQR is the same as the difference between medians.
Therefore, the median score of the seventh-grade class is 2 points greater than the median score of the sixth-grade class. The difference is the same as the IQR
Hence, the correct answer is option B
What are the solutions of x^2=-7x-8
Answers
The solutions to the quadratic equation x² = -7x - 8 are x equals \(\frac{-7 - \sqrt{17}}{2}, \frac{-7 + \sqrt{17}}{2}\).
What are the solutions to the quadratic equation?Given the quadratic equation in the question:
x² = -7x - 8
To find the solutions of the quadratic equation x² = -7x - 8, we can rearrange it into standard quadratic form, which is ax² + bx + c = 0, and apply the quadratic formula.
x² = -7x - 8
x² + 7x + 8 = 0
a = 1, b = 7 and c = 8
Plug these into the quadratic formula: ±
\(x = \frac{-b \± \sqrt{b^2 -4(ac)}}{2a} \\\\x = \frac{-7 \± \sqrt{7^2 -4(1*8)}}{2*1} \\\\x = \frac{-7 \± \sqrt{49 -4(8)}}{2} \\\\x = \frac{-7 \± \sqrt{49 - 32}}{2} \\\\x = \frac{-7 \± \sqrt{17}}{2} \\\\x = \frac{-7 - \sqrt{17}}{2}, \frac{-7 + \sqrt{17}}{2}\)
Therefore, the values of x are \(\frac{-7 - \sqrt{17}}{2}, \frac{-7 + \sqrt{17}}{2}\).
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What happens to the graph of f(x) when it is multiplied
by a number between - 1 and 0?
Answers
Prove that 3a + 3b is equivalent to 3(a + b) when a = 4 and b = 3.
Answers
Step-by-step explanation:
= 3a+3b
or, 3×4+3×3
or, 12+9
or, 21
again,
= 3(a+b)
or, 3×(4+3)
or, 3×7
or, 21
hence 3a+3b is equivalent to 3(a+b) is proven
Write an explicit formula for an, the nth term of the sequence 23, 19, 15, ....
Answers
Answer:
\(a_{n}\) = - 4n + 27
Step-by-step explanation:
there is a common difference between consecutive terms, that is
19 - 23 = 15 - 19 = - 4
this indicates the sequence is arithmetic with nth term
\(a_{n}\) = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
here a₁ = 23 and d = - 4 , then
\(a_{n}\) = 23 - 4(n - 1) = 23 - 4n + 4 = - 4n + 27 or 27 - 4n
what is the value of the power a if 5a = 1/125?
Answers
Answer:
If
5
a
=
1
125
, find the value of a?
Step-by-step explanation:
There is a base f
5
on the left side, so write the denominator on the right as a base of
5
as well.
5
a
=
1
125
5
a
=
1
5
3
Move the base to the numerator.
5
a
=
5
−
3
If the bases are equal, the indices are equal:
∴
a
=
−
3
Consider the following current information for Galaxy Inc::
Output = 200 units
ATC = $50
What is the total cost of producing 200 units of output?
a. $10,000
b. $8,000
c. $1,100
d. Non
Answers
The answer is (a) $10,000.
How the total cost of producing 200 units of output can be found?The total cost of producing 200 units of output can be found by multiplying the output (200 units) by the average total cost (ATC) per unit, which is given as $50. Therefore, the total cost is:
Total Cost = Output x ATC
Total Cost = 200 x $50
Total Cost = $10,000
Therefore, the answer is (a) $10,000.
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Suppose two students from Georgia State University, working as interns for Select one answer the American National Elections Studies agency (ANES), are both 10 points assigned to survey a random sample of registered voters and ask whether or not they will vote for a certain candidate. The first intern plans to select 500 voters and the second intern plans to select 1500 voters. If each intern conducted the study repeatedly selecting different samples of people each time... but using the same sample size), which one of the following would be true regarding the resulting sample proportion, p, of "yes" responses for each intern? A. For either sample size, using the same size each time, as long as the sampling is done with replacement, their mean would be o. B. Different sample proportions, p, would result for each intern, but for either sample size, they would be centered (have their mean) at the true population proportion, P. C. Different sample proportions, p, would result for each intern, but for the intern using a sample size of 1500 they would be centered (have their mean) at the true population proportion, P, whereas for sample size 500 they would not. D. Different sample proportions, p, would result for each intern, but for sample size 500 they would be centered (have their mean) at the true population proportion, P, whereas for sample size 1500 they would not.
Answers
Answer: Different sample proportions, ^p, could result for each intern, but for either sample size, they would be centered (have their mean) at the true population proportion, p.
Step-by-step explanation:
I need help writing this in simplest form. What is the square root of 16/81 in simplest form ?
Answers
We will have the following:
\(\sqrt[]{\frac{16}{81}}=\frac{\sqrt[]{16}}{\sqrt[]{81}}=\frac{4}{9}\)So, the expression in its simplest form is 4/9.
***Explanation***
First we are given the expression:
\(\sqrt[]{\frac{16}{81}}\)From the properties of roots, we will have that if a root contains a fraction, then we can rewrite the expression applying the root to the numerator and denominator [Separately], this is:
\(\sqrt[]{\frac{16}{81}}=\frac{\sqrt[]{16}}{\sqrt[]{81}}\)Since, these expressions are equivalent, then we solve for each one, that is:
\(\begin{cases}\sqrt[]{16}=4 \\ \& \\ \sqrt[]{81}=9\end{cases}\)Then, we simply replace in the fraction form:
\(\sqrt[]{\frac{16}{81}}=\frac{4}{9}\).Mobile banner ads perform significantly better than desktop banners.
False or true?
Answers
It is false that mobile banner ads perform significantly better than desktop banners.
There is no clear consensus on whether mobile banner ads or desktop banner ads perform better. The effectiveness of banner ads depends on various factors such as the placement of the ad, its design, and the target audience.
However, it is true that mobile usage has been increasing rapidly in recent years, and more people are accessing the internet through their mobile devices than through desktop computers. Therefore, it is important for advertisers to optimize their ads for mobile devices and ensure that they are mobile-responsive.
Nevertheless, it cannot be generalized that mobile banner ads are more effective than desktop banner ads. The effectiveness of an ad should be evaluated on a case-by-case basis, taking into account the specific objectives, target audience, and design of the ad.
Therefore, it is important for advertisers to test their banner ads on both desktop and mobile devices to determine which platform works best for their specific campaign. And the given statement is false.
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Given the points B(-2, 10) and C(8, 7), find one point D that makes both statements true.
The slope of the line connecting B and D is 2. The slope of the line connecting C and D is - 3.
Answers
Answer:
D(3.4, 20.8)
Step-by-step explanation:
Given points B(-2, 10) and C(8,7), you want the coordinates of point D such that BD has a slope of 2 and CD has a slope of -3.
SlopeThe equation for the slope between two points is ...
m = (y2 -y1)/(x2 -x1)
Using this formula and the given points, we can find D(x, y) to satisfy ...
2 = (y -10)/(x +2)
-3 = (y -7)/(x -8)
SolutionPutting each of these equations in general form, we have ...
2(x +2) -(y -10) = 0
2x -y +14 = 0
and
-3(x -8) -(y -7) = 0
3x +y -31 = 0
Adding the two equations eliminates the y-variable:
(2x -y +14) +(3x +y -31) = 0
5x -17 = 0 . . . . . . simplify
x -3.4 = 0 . . . . . . divide by 5
x = 3.4 . . . . . . . . . add 3.4
Substituting for x in the first equation gives ...
2(3.4) -y +14 = 0
y = 6.8 +14 = 20.8 . . . . add y
The point D has coordinates (3.4, 20.8).
__
Additional comment
The graph shows the equations of the lines written in point-slope form.
HELLp NOWWEEE I really need it plsss
Answers
Answer: diameter is 5.2 radius is 2.6
Step-by-step explanation:
Write an exponential function y= ab^x for a graph that includes (-1,12.5) and (4,4.096)
Answers
An exponential function for a graph that includes (-1,12.5) and (4,4.096) is,
⇒ y= 10(0.80)ˣ
What is Function?A relation between a set of inputs having one output each is called a function.
Given that;
An exponential function is,
⇒ y= abˣ
Here, An exponential function for a graph includes (-1,12.5) and (4,4.096).
Hence, The points must satisfy the exponential function.
So, Put x = - 1, y = 12.5,
⇒ y = abˣ
⇒ 12.5 = ab⁻¹
⇒ 12.5b = a .. (i)
And, Put x = 4, y = 4.096,
⇒ 4.096 = ab⁴
⇒ 4.096 = ab⁴
From (i), put the value of a in above equation we get;
⇒ 4.096 = (12.25b) b⁴
⇒ 4.096 = 12.25b⁵
⇒ 4.096 /12.25 = b⁵
⇒ b = ⁵√0.33
⇒ b = 0.80
And, from (i);
⇒ 12.5b = a
⇒ 12.5 × 0.80 = a
⇒ a = 10
Hence, An exponential function for a graph that includes (-1,12.5) and (4,4.096) is,
⇒ y= abˣ
⇒ y= 10(0.80)ˣ
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The scores earned in a flower-growing competition are represented in the stem-and-leaf plot.
2 0, 1, 3, 5, 7
3 2, 5, 7, 9
4
5 1
6 5
Key: 2|7 means 27
What is the appropriate measure of variability for the data shown, and what is its value?
The IQR is the best measure of variability, and it equals 16.
The range is the best measure of variability, and it equals 45.
The IQR is the best measure of variability, and it equals 45.
The range is the best measure of variability, and it equals 16.
Answers
The appropriate measure of variability for the given data is the IQR, and its value is 16.
Based on the given stem-and-leaf plot, which represents the scores earned in a flower-growing competition, we can determine the appropriate measure of variability for the data.
The stem-and-leaf plot shows the individual scores, and to measure the spread or variability of the data, we have two commonly used measures: the range and the interquartile range (IQR).
The range is calculated by subtracting the smallest value from the largest value in the dataset. In this case, the smallest value is 20, and the largest value is 65. Therefore, the range is 65 - 20 = 45.
The interquartile range (IQR) is a measure of the spread of the middle 50% of the data. It is calculated by subtracting the first quartile (Q1) from the third quartile (Q3). Looking at the stem-and-leaf plot, we can identify the quartiles. The first quartile (Q1) is 25, and the third quartile (Q3) is 41. Therefore, the IQR is 41 - 25 = 16.
In this case, both the range and the IQR are measures of variability, but the IQR is generally preferred when there are potential outliers in the data. It focuses on the central portion of the dataset and is less affected by extreme values. Therefore, the appropriate measure of variability for the given data is the IQR, and its value is 16.
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Solve the following system of equations:
x + 3y = −4
x + 5y = −6
(1, 1)
(−1, 1)
(1, −1)
(−1, −1)
Answers
Answer:
(-1,-1)
thats where they intersect
and thats the answer
Step-by-step explanation:
plz give brainlyest
Answer:
(-1,-1)
Step-by-step explanation:
.-50 + d = 2c, for c
Answers
Answer:
-50+d=2c
2c=d-50
2c/2=d-50/2
c=1/2d-25
Step-by-step explanation:
Holly opened a savings account at a local bank. She deposited $3,100.00 into the account 9 years ago. If the account earns an
annual simple interest rate of 7.1%, how much interest has she earned?
Answers
Answer:
She has earned $1980.90 in interest.
Step-by-step explanation:
Since we know that she deposited $3100 9 years ago, we know we have to find the amount of interest she gets per year times 9. So $3100*0.071 = $220.10 and then $220.1*9 = $1980.90. Therefore, she has earned $1980.90 in interest.
Answer:
The answer is 1980.9
Step-by-step explanation:
Formula is PxRxT
P=Princlple(Amount of money desposited)
R=Rate the rate in deciaml form. Example: 7.1=0.071
T=Time
SI=3100x0.071x9=1980.9
P R T
a plane left Atlanta at 11:30 a.m and flew to an airport near boston. the plane was due at boston at 3:15 pm
Answers
The elapsed time during the trip is 4 hours 10 minutes.
With that in mind,
the plane left Atlanta at 11:30 am for an airport near Boston. The plane was scheduled to arrive in Boston at 3:15 pm.
Mathematics is about a number of operations based on information.
because the plane leaves Atlanta at 11:30 am and he lands at Boston airport 25 minutes later than the actual arrival time.
The arrival time was 3:15 PM, but due to a delay, the plane landed at Boston Airport at 3:40 PM. 15:40 is 15:40 on a 24-hour clock. So
Elapsed Time = 15:40 - 11:30
= 4 : 10
means 4 hours and 10 minutes
So the elapsed time during the trip is 4 hours and 10 minutes.
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What is the missing exponent? in numerical answer.
Answers
Answer:
x la 2
Step-by-step explanation:
when we multiply with powers the powers gather
Domain and Range The function f(x) is defined as a set or ordered pairs. Using that set of data, identify the Domain and Range of f(x). Write your answer as an ordered list enclosed in curly brackets. f(x) = {(-19, 12), (-17,90), (-7,2), (2,76), (3, 23), (34, 87)} Domain: Range:
Answers
The domain of the function f(x) is {-19, -17, -7, 2, 3, 34}, which represents the set of all possible x-values or inputs for the function. The range of the function f(x) is {2, 12, 23, 76, 87, 90}, which represents the set of all corresponding y-values or outputs for the function.
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In the given set of ordered pairs, the x-values are -19, -17, -7, 2, 3, and 34. Therefore, the domain of the function f(x) is {-19, -17, -7, 2, 3, 34}.
On the other hand, the range of a function represents the set of all possible output values (y-values) that the function can produce. Looking at the given set of ordered pairs, the corresponding y-values are 12, 90, 2, 76, 23, and 87. Thus, the range of the function f(x) is {2, 12, 23, 76, 87, 90}.
In summary, the domain of the function f(x) consists of the x-values {-19, -17, -7, 2, 3, 34}, and the range of the function includes the corresponding y-values {2, 12, 23, 76, 87, 90}. These sets represent the valid inputs and outputs of the function based on the given set of ordered pairs.
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Plot the points in a coordinate plane and sketch ∠ XYZ. Then classify it as right, acute, or obtuse.
X(5,-3), Y(4,-1), Z(6,-2)
Answers
The points X(5,-3), Y(4,-1) and Z(6,-2) can be plotted as given below. From the obtained graph ∠XYZ is the obtuse angle.
The given coordinates are X(5,-3), Y(4,-1) and Z(6,-2).
We need to classify ∠XYZ whether it is right, acute, or obtuse.
How to plot the points in a cartesian plane?A cartesian plane or coordinate plane can be defined as a plane formed by the intersection of two coordinate axes that are perpendicular to each other. The horizontal axis is called the x-axis and the vertical one is the y-axis. These axes intersect with each other at the origin whose location is given as (0, 0). Any point on the cartesian plane is represented in the form of (x, y). Here, x is the distance of the point from the y-axis and y is the distance from the x-axis.
Now, X(5,-3), Y(4,-1) and Z(6,-2) can be plotted as given below:
The given points X(5,-3), Y(4,-1) and Z(6,-2) have positive x-coordinate and negative y-coordinate.
In the cartesian plane, the fourth quadrant positive x-coordinate and negative y-coordinate.
So, all points X(5,-3), Y(4,-1) and Z(6,-2) lie in the fourth quadrant.
From the graph, we can see that the two lines make an obtuse angle, which is more than 180° and less than 360°.
From the obtained graph ∠XYZ is the obtuse angle.
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How do you find the largest and smallest number in an array of 1 100?
Answers
The simplest solution for this problem is to traverse the entire array and store the minimum and maximum values.
C++ implementation for the following problem
#include <iostream>
int main(){
int arr[] = {1,2,3,4,5,6,7,8,9...........................99,100}; //initialize array
int min = arr[0]; //start value
int max= arr[0]; //start value
for(int i:arr) {
if(i<min) min=i; // if current number lower update min
if(i>max) max=i; // if current number bigger update max
}
std::cout<<"Min Value: "<<min<<"\nMax Value: "<<max<<"\n";
return 0;
}
The simplest solution for this problem is to traverse the entire array and store the minimum and maximum values.
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The Smith family paid $89,250 for their home. The value of the home increases4.5% each year.Create an equation that represents the value, v, of the house after t years.
Answers
We have the following:
In this case we would have a multiplication between the initial value and 100% added to the increase each year
\(v=89250\cdot(1+0.045\cdot t)\)a biologist wants to calculate the number of fish in a lake. on may 1 she catches a random sample of 60 fish, tags them, and releases them. on september 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. to calculate the number of fish in the lake on may 1, she assumes that 25% of these fish are no longer in the lake on september 1 (because of death and emigrations), that 40% of the fish were not in the lake may 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the september 1 sample are representative of the total population. what does the biologist calculate for the number of fish in the lake on may 1?
Answers
The total number of fishes present in the lake on my 1 is 840.
Explain the term percentage of the number?A figure or ratio that may be stated as a percentage of 100 is a percentage. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100.The given data for the question-
42 fish were present in May out of the 70 fish that were caught in September, 40% of which were absent in May. Noting that the 25% death rate has no impact on the answer since both tagged and untagged fish die, the percentage the tagged fish inSeptember is equivalent to a percentage of tagged fish in May.
3/42 = 60/x
x = 60 *42 / 3
x = 840
Thus, the total number of fishes present in the lake on my 1 is 840.
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A holiday store is having a sale on bows and rolls of wrapping paper. One sign in the store states, "Buy 1 bow and 1 roll of wrapping paper for only $6. 0. " A second sign in the store states, "5 bows and 1 roll of wrapping paper will only cost $10. 0. "
Answers
When a store advertises sales on different combinations of goods, they are engaging in price discrimination. In this case, the holiday store is practicing second-degree price discrimination because it has different prices for different quantities of bows and wrapping paper.
According to the signs in the store, there are two different deals on bows and rolls of wrapping paper. One sign says that one bow and one roll of wrapping paper can be purchased for $6.00. This means that the store is offering a bundle deal that gives a discount if you purchase both items together. The second sign offers five bows and one roll of wrapping paper for $10.00.
This means that the store is offering an even larger discount for customers who purchase a larger quantity of bows.The fact that the store is offering different prices based on the quantity of goods purchased is an example of price discrimination. Specifically, the store is practicing second-degree price discrimination because it is offering different prices for different quantities of goods.
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a student measures the mass of a rock three times using the same properly calibrated analog (not digital) balance and gets slightly different values: 22.46g, 22.42g, and 22.44g. what type of error may have occurred here (systematic and/or random)? how would the student minimize the impact of this error during data analysis?
Answers
The errors that may have occurred in the student's measurements are random errors. These types of errors can arise from various sources, such as slight variations in measurement techniques or environmental factors.
Since the measured values are slightly different, it suggests that random errors affected the measurements rather than systematic errors, which would result in consistently incorrect measurements.
To minimize the impact of random errors during data analysis, the student can calculate the average of the three measured values. The average value can serve as the best estimate of the true value of the mass of the rock. Additionally, the student can calculate the standard deviation of the measured values to determine the precision of the measurements. The smaller the standard deviation, the more precise the measurements.
The student can further minimize random errors by taking multiple measurements and averaging them to obtain a more accurate value. Additionally, using a digital balance instead of an analog one may reduce random errors as digital balances have greater precision and accuracy.
It is also important for the student to use proper measurement techniques such as ensuring that the balance is calibrated and zeroed correctly before each measurement. The student should also avoid applying excessive force on the balance and ensure that the rock is stable and level during measurement.
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