The general solution of the given equation is y² = 8x²Inx + cx².
Linear first order differential equation:A first-order linear differential equation is one that has the formula:
[tex]\frac{d y}{d x}+p(x) y=q(x)[/tex]
where p(x) and q(x) are continuous functions of x.
We begin by determining the integrating factor for the first-order linear differential equation.
I.F. = [tex]e^{\int p(x) d x}[/tex]
The solution to the first order linear differential equation is:
[tex]y \cdot e^{\int p(x) d x}=\int q(x) \cdot e^{\int p(x) d x} d x+c .[/tex]
Where c is the constant of integration.The solution to the first order linear differential equation is:
[tex]\int \frac{1}{x} d x=\ln x+c[/tex]
Where c is the constant of integration.So, given: [tex]y^{\prime}=\frac{\left(4 x^2+y^2\right)}{(x y)}[/tex]
The above differential equation can be written as:
[tex]\begin{aligned}&\frac{d y}{d x}=4 \frac{x}{y}+\frac{y}{x} \\&\frac{d y}{d x}-\frac{y}{x}=\frac{4 x}{y} \\&y \frac{d y}{d x}-\frac{y^2}{x}=4 x\end{aligned}[/tex]
Substitute, [tex]y^2=t, 2 y \frac{d y}{d x}=\frac{d t}{d x}[/tex].
[tex]\begin{aligned}&\frac{1}{2} \frac{d t}{d x}-\frac{t}{x}=4 x \\&\frac{d t}{d x}-\frac{2 t}{x}=8 x\end{aligned}[/tex]
Because this is a first-order linear differential equation, we must first determine the integrating factor:
[tex]\text { I. F. }=e^{-2 \int \frac{1}{x} d x}=e^{-2 \ln x}\\\text { I.F. } F=\frac{1}{x^2}[/tex]
As a result, the following is the general solution to the given IVP:
[tex]$$t \cdot(I . F .)=\int(I . F .) \cdot 8 x d x+c$$$t \cdot\left(\frac{1}{x^2}\right)=\int\left(\frac{1}{x^2}\right) \cdot 8 x d x+c$$t \cdot\left(\frac{1}{x^2}\right)=8 \int \frac{1}{x} d x+c$$t \cdot\left(\frac{1}{x^2}\right)=8 \ln x+c$$t=8 x^2 \ln x+c x^2$[/tex]
By reversing the value of t:
[tex]y^2=8 x^2 \ln x+c x^2[/tex]
We now test the solution by substituting [tex]y^2=8 x^2 \ln x+c x^2[/tex] it in the original differential equation.
[tex]\begin{aligned}&\frac{d}{d x}\left(y^2\right)=\frac{d}{d x}\left(8 x^2 \ln x+c x^2\right) \\&2 y \frac{d y}{d x}=16 x \ln x+8 x+2 c x \\&\frac{d y}{d x}=\frac{16 x \ln x+8 x+2 c x}{2 y} \\&\frac{d y}{d x}=\frac{8 x \ln x+4 x+c x}{y}\end{aligned}[/tex]
We calculated the value of [tex]y^2[/tex] and [tex]\frac{d y}{d x}[/tex] from the original differential equation:
[tex]\begin{aligned}&\frac{8 x \ln x+4 x+c x}{y}=\frac{4 x^2+8 x^2 \ln x+c x^2}{x y} \\&8 x \ln x+4 x+c x=8 x \ln x+4 x+c x .\end{aligned}[/tex]
Therefore, the general solution of the given equation is y² = 8x²Inx + cx².
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The complete question is given below:
For the following ODE, find a general solution, show the steps of derivation and check your answer by substitution.
[tex]y^{\prime}=\frac{\left(4 x^2+y^2\right)}{(x y)}[/tex]
a) Work out the value of 23 +52
b) Work out the value of 72-33
c) Work out the value of 82 + √100
Answer:
Step-by-step explanation:
One
23 + 52 = 75 This can be done using columns
2 3
5 2 No carries Add.
7 5
Two
72 - 33 = 39
7 2
3 3 You have to borrow a ten from the 7 to make this work
3 9
Three
82 + sqrt(100)
sqrt(100) = 10
82 + 10 = 92
Which of the following could be used to
determine the number of hits a player has
during the season?
h = Ab
h=A+b
A.
B.
C. h=
D. h=
A
b
b
A
It was mostly during C . Dry season
The “Cricket Season” is the period of days where cricket is played in a country when the pitches get dry and also this is the period where the domestic matches begin.
And this season varies with countries, in India it starts with the Ranjit Trophy, Sheffield Shield for Australia etc.
Cricket seasons are the times of the year when cricket is played. Because top-class cricket is almost always played outdoors, on uncovered pitches, and rain prevents play, the seasons in each country are geared to coincide with the driest months of the year
The cricket season starts at the beginning of April and finishes at the end of September. As the time zones move around the planet, the seasons will differ in other countries.
In the UK,
professional games have occasionally taken place at the end of March but the season will normally begin at the start of April.
In India
The cricket season in India will usually begin in mid-September. Often, there are senior women’s games starting things off before men’s tournaments commence in October.
In Australia
Cricketers in Australia may start playing in September, especially in the west of the country which gets the warmer weather first.
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Show the correct steps and solution
After simplifying the expression, - ( 7 x - 3 y ) + 2 ( 3/4 x + 1/2 y), we get the result as - 11 / 2 x + 4 y.
- ( 7 x - 3 y ) + 2 ( 3/4 x + 1/2 y)
First we will multiply 2 and open the brackets.
So, we get that:
- 7 x + 3 y + 3/2 x + y
Combining the like terms, we get that:
= ( - 14 + 3 ) x / 2 + 3 y + y
Simplifying and solving the expression to obtain the result, we get that:
= - 11 / 2 x + 4 y.
Therefore, after simplifying the expression, - ( 7 x - 3 y ) + 2 ( 3/4 x + 1/2 y), we get the result as - 11 / 2 x + 4 y.
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I need help with these
Irrational because the number keeps going on forever and it does not repeat a certain pattern like 0.24242424...
What is the domain of the graph?
given that 23x-32y =o where x and y are number bases express x in terms of o and y
Expressing x in terms of 0 and y gives x = 0 + (32/23)y
Given the expression 23x-32y = 0 with number bases x and y.
This equation actually expresses 0 in terms of x and y as 0 is isolated to one side of the equality.
We have to express x in the same equation in terms of 0 and y.
So we need to isolate x to one side of the equality.
23x-32y = 0
⇒23x = 0 +32y
Dividing throughout by 23 gives,
⇒ x = 0 + (32/23)y
This is x expressed in terms of 0 and y
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Which is more affected by extreme observations, the mean or median? and how about the standard deviation or iqr?
Mean and standard deviation is more affected by extreme observations.
Mean is the summation of all the values divided by the total number of values. It is also called as average. Median is the middle value.Standard deviation is the dispersion of a dataset relative to its mean. In case of extreme items arithmetic mean is affected very much. For this reason, it no longer remains representative of the distribution. As standard deviation is the positive square root of the arithmetic mean of the square of deviations of all given values, it is also affected by extreme items. As median is a partition value, it is not affected by extreme items.
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Atsu and Sabu shared an amount of money in the ratio 3:7 respectively. if Sabu received 20 cedis more than Atsu, how much was shared
Answer:
Step-by-step explanation:
Atsu : Sabu
3 : 7
x : x + 20
x is equal to the number of cedis
cross multiply the ratio above
3(x+20) = 7x
3x + (20×3) = 7x
3x + 60 = 7x
3x - 3x + 60 = 7x - 3x (put - 3x on both sides of the equation)
60 = 4x
60/4 = 4x/4 (divide both sides by 4)
15 = x
Factor completely:
4x2−144
A
4(x2−36)
B
4(x+6)(x−6)
C
(2x−12)(2x+12)
D
(x+12)(4x−12)
Answer:
C
Step-by-step explanation:
(2x - 12)(2x + 12)
4x² + 24x - 24x - 144
4x² - 144
The sum of three consecutive integers is -27. Find the value of the middle of the
three.
Solving the Question
Let the middle integer be x.
We're given:
[tex](x-1) + x + (x+1) = -27[/tex]Solve for x.[tex](x-1) + x + (x+1) = -27[/tex]
Open up the brackets:
[tex]x-1 + x + x+1 = -27[/tex]
Combine like terms:
[tex]3x = -27[/tex]
Divide both sides by 3:
[tex]\dfrac{3x}{3} = \dfrac{-27}{3}[/tex]
[tex]x=-9[/tex]
AnswerThe value of the middle integer is -9.
A+company+packed+8+dark+chocolates+in+a+box+of+32+assorted+chocolates.+which+of+these+does+not+represent+inferential+statistics?+a.)+25%+of+the+chocol
Answer:
it is 3
Step-by-step explanation:
beacuse i did it
A circular plate has a circumference of 34.54in. What is the area? Use 3.14 for pie and round to the nearest square inch.
The area of the circular plate is 9234 inches²
How to find area of the circular plate?The plate is circular and have a circumference of 34.54 inches.
Hence, let's find the radius of the circular plate using the circumference of the circular plate.
circumference of the circular plate = 2πr
where
r = radiusTherefore,
34.54 = 2π × r
r = 34.54 / 2π
r = 17.27π
Therefore,
area of the circular plate = πr²
area of the circular plate = π × ( 17.27π)²
area of the circular plate = π³ × 298.2529
area of the circular plate = 3.14³ × 298.2529
area of the circular plate = 9233.65447952
area of the circular plate = 9234 inches²
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The function m(x) = |x-441 represents the absolute deviation (in years) of a man's agex (in years) from the average age of men in a
city. The function w(x)=x-47| represents the absolute deviation (in years) of a woman's age x (in years) from the average age of
women in the same city. Compare the absolute deviations for a man and a woman who are both 64 years old.
A
The age of the man is years from the average age of men in the city. The age of the woman isyears from the average age of women in the city.
the absolute deviation for the man is 3 years more than the one for the woman.
How to compare the deviations?We know that the absolute value function:
m(x) = |x - 44|
Represents the absolute deviation of man's age, in years.
This means that the average age of a man in that city is 44 years old, so if a man has 64 years, then the absolute deviation is given by:
m(64) = |64 - 44| = 20
The absolute deviation for a man of 64 years old living in this city is 20 years.
Similar for the case of women, the equation here is:
w(x) = |x - 47|
If a woman has 64 years old, then the absolute deviation is:
w(64) = |64 - 47| = 17
The absolute deviation is 17 years.
We can see that the absolute deviation for the man is 3 years more than the one for the woman.
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How can you rearrange two digits in the number 2,957,648 so that the value of the digit 4 is 10 times greater?
Answer:
2,957,468
Step-by-step explanation:
Each digit to the left is worth 10 times more than the same digit one place to the right.
If 2,957,648 becomes 2,957,468, the 4 moves from the tens place to the hundreds place, and now its value is 400 instead of 40, so it's 10 times greater.
solve the equation tanx+secx=2 given that x is greater than or equal to 0 and less than or equal to 2pie
X = 0.6444
Trigonometric Identities are equality statements that hold true for all values of the variables in the equation and that use trigonometry functions.
There are numerous distinctive trigonometric identities that relate a triangle's side length and angle. Only the right-angle triangle is consistent with the trigonometric identities.
The six trigonometric ratios serve as the foundation for all trigonometric identities. Sine, cosine, tangent, cosecant, secant, and cotangent are some of their names. The adjacent side, opposite side, and hypotenuse side of the right triangle are used to define each of these trigonometric ratios. The six trigonometric ratios are the source of all fundamental trigonometric identities.
tan x +sec x =2
⇒sin x /cosx + 1/ cos x = 2
⇒1+sin x / cos x =2
⇒ (sin² x/2 +cos ² x/2 +2 sin x/2 cos x/2 )/( cos ² x/2 - sin²x/2 ) = 2
⇒(cos x/2 +sin x/2)² / ( cos ² x/2 - sin²x/2 ) = 2
⇒( cos x/2+sin x/2)² / (cos x/2 + sin x/2)(cos x/2-sinx/2) = 2
⇒(cos x/2 +sin x/2)/ (cos x/2-sin x/2) = 2
Dividing by cos x/2 in both numerator and denominator,
⇒(1+tan x/2)/(1-tan x/2) =2
⇒ 1 + tan x/2 = 2 - 2tanx/2
⇒ tan x/2 = 1/3
⇒ x= 0.6444
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________ is concerned with making predictions about a population from analyzing a sample or through observations.
Inferential statistic is concerned with making prediction about a population from analyzing a sample or through observation.
Here,
We have to find the concerned with making prediction about a population from analyzing a sample or through observation.
What is Inferential statistic?
Inferential statistics is used in measurements from the sample of subjects in the experiment to compare treatment groups and to make generalizations about the larger population of subjects.
Now,
Inferential statistic is concerned with making prediction about a population from analyzing a sample or through observation.
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In a normal distribution the theoretical mean is 30 and the theoretical standard deviation is 5. Find: a) the area below 24, b) the area between 24 and 36, c) the point that has 95% of the area below it.
Answer:
a) P(X < 24) = 0.1151
b) P(24 < X < 36) = 0.7698
c) X value for P(x < X) = 0.95 is X = 38.225
Step-by-step explanation:
a) P(X < 24)
Since μ = 30 and σ = 5 we have:
[tex]\text {z value = } \dfrac{24-30}{5}} = -1.2[/tex]
P(X < 24) = P(Z < -1.2) = 0.1151
You can use a calculator or the standard normal tables to determine P(Z < -1.2)
b) P(24 < X < 36)
First find P(X < 36) using the technique detailed in part a)
For 36, the Z value is
[tex]\dfrac{36-30}{5} = 1.2[/tex]
P(X < 36) = P(z < 1.2) = 0.8849
P(24 < X < 36) = P(X < 36) - P(X < 24) = 0.8849 - 0.1151 = 0.7698
c) X value below which 95% of the area is covered
Using a calculator (or tables) the z value corresponding to P(z < Z) = 0.95 is Z = 1.65(approx)
[tex]\text{We have } \dfrac{X - \mu}{\sigma} = z\\\\\dfrac{X - 30}{5} = 1.645\\\\\text{Multiplying both sides by 5: }\\\\X - 30 = 1.645 \times 5 = 8.225\\\\X = 30 + 8.225 = 38.225[/tex]
So X value is 38.225-
O Solve questions Pg #61 Prá
1 Correct 78. 4695 to
a) 1 decimal place.
c) the nearest hundredth
The number corrected to 1 decimal place is 78.5
The number corrected to the nearest hundredth is 78.47
EstimationFrom the question, we are to estimate the given number to the nearest 1 decimal place
The given number is
78.4695
Correcting to 1 decimal place
In the given number, we can observe that the hundredth digit is greater than 5, thus we will add 1 to the tenth digit
Thus,
78.4695 corrected to 1 decimal place is 78.5
Correcting to the nearest hundredth
The thousand digit is greater than 5, thus we will add 1 to the hundredth digit
Thus,
78.4695 corrected to the nearest hundredth is 78.47
Hence,
The number corrected to 1 decimal place is 78.5
The number corrected to the nearest hundredth is 78.47
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A number Q is at least -2 and less than 6
Answer:
Q=-8
Step-by-step explanation:
Q-2-6
Hans spent $38 on fruit at the grocery store. He spent a total of $40 at the store. What percentage of the total did he spend on fruit?
MODELING REAL LIFE Bowling alley A charges $3.75 to rent shoes and $4 per game. Bowling alley B charges
$2.50 to rent shoes and $4.50 per game.
a. For what numbers of games is the total cost, including a pair of rental shoes, less at bowling alley A? at bowling
alley B?
Bowling alley A:__ or more games
Bowling alley B:__
or __
games
b. Bowling alley A increases the cost per game by $0.50. How does this affect your answer in part (a)?Explain.
Using linear functions, we have that:
a. For 6 or more games, the cost a bowling alley A is less, and for 4 or less games, the cost at bowling alley B is less.
b. For any number of games, the cost at alley B will be less.
What is a linear function?
A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.For the functions in this problem, we have that:
The slope is the cost per game.The intercept is the cost of 2 shoes.Hence they are given as follows:
A(x) = 4x + 7.5.B(x) = 4.5x + 5.The cost for A will be less when:
A(x) < B(x).
Hence:
4x + 7.5 < 4.5x + 5
-0.5x < -2.5
0.5x > 2.5
x > 2.5/0.5
x > 5.
Hence for 6 or more games, the cost a bowling alley A is less, and for 4 or less games, the cost at bowling alley B is less.
For item b, with the increase of $0.5 per game, we have that:
A(x) = 4.5x + 7.5.
Then the costs per game will be the same, while alley B has the lower intercept, meaning that for any number of games, the cost at alley B will be less.
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A paper company needs to ship paper to a large printing business. The paper will be shipped in small boxes and large boxes. Each small box of paper weighs 45 pounds and each large box of paper weighs 75 pounds. There were 3 more small boxes shipped than large boxes and the total weight of all boxes was 1455 pounds. Write a system of equations that could be used to determine the number of small boxes shipped and the number of large boxes shipped. Define the variables that you use to write the system
Answer: The equations are:
X+3=Y;
45X + 75Y = 1455 ;
The variables used are X and Y specifying number of small and large boxes
Step-by-step explanation:
Let X be the number of small boxes
Let Y be the number of big boxes
The first equation is: X+3=Y; ( Since there were 3 more small boxes shipped than large boxes )
The second equation is: 45X + 75Y = 1455 ( this is because the small box weighs 45 pounds and big box weighs 75 pounds. The total weight of the small boxes is 45 times the weight of each individual box. And the total weight of the second boxes is equal to 75Y.)
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Answer:
Step-by-step explanation:
The variables used are A and B specifying number of small and large boxes
Let the number of small box is A
Let the number of large box is B
The first equation is: A+3=B;
( Since there were 3 more small boxes shipped than large boxes )
The second equation is: 45A + 75B = 1455 ( this is because the small box weighs 45 pounds and large box weighs 75 pounds. The total weight of the small boxes is 45 times the weight of each box. And the total weight of the second boxes is equal to 75B.)
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5x - 8x - 7=-3x+5-7 won’t pull an answer to check my answer
What percentage of the data values are greater than 80?
64.71% percent of the data values are greater than 80.
What is the percentage?
In Ancient Rome, long before the actuality of the decimal system, computations were constantly made in fractions in the multiples of '/ 100', For illustration, Augustus levied a duty of1/ 100 on goods sold at sale known as centesimal Rerum valium. Computation with these fractions was original to calculating chances. They named it as a percentage.As nomenclatures of capitalism grew in the Middle ages, computations with a denominator of 100 came increasingly standard, analogous that from the late 15th century to the early 16th century, it came common for calculation-handbooks to include analogous computations known as percentage.Multitudinous of these handbooks applied these styles to profit and loss, interest rates, and the Rule of Three. By the 17th(seventeenth) century, it was standard to quote interest rates in hundredthsMin. value = 30
Max. value = 85
Percentage (%) of data values greater than 30(thirty) = [tex]\frac{55}{85} X 100[/tex]
= 64.71%
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Given f(x) = 2x² - 5x, evaluate f(x-4) + f(x)
Step-by-step explanation:
f(x) = 2x² - 5x
= f(x-4) + f(x)
= [2.(x-4)² - 5.(x-4)] + [2x² - 5x]
= [2.(x² - 8x + 16) - 5x + 20] + [2x² - 5x]
= [2x² - 16x + 32 - 5x + 20] + [2x² - 5x]
= [2x² - 21x + 52] + [2x² - 5x]
= 4x² - 26x + 52
Antwan determines the distance between the points –7 and 2 on a number line. Maggie determines the difference between the numbers –7 and 2. How are Antwan’s and Maggie’s solutions related?
Maggie’s solution is the absolute value of Antwan’s solution.
Antwan’s solution is the absolute value of Maggie’s solution.
Both solutions are greater than either of the two numbers in the problem.
Both solutions are less than either of the two numbers in the problem.
Antawan's solution is the absolute value of Maggie's solution is the solution related to Antawan's and Maggie's solution.
Given
Antawan determines the distance between two points as 9.
Maggie finds the difference between two numbers ₋ 7 ₋ 2 = ₋9
Hence we notice that Antawan's solution is the absolute value to the Maggie's solution.
The non-negative value of x, regardless of its sign, is the absolute value (or modulus) | x | of a real number x. For instance, 5 has an absolute value of 5 and so does 5, which likewise has an absolute value of 5. One way to conceptualize a number's absolute value is as its separation from zero on the real number line.
hence option 2 is right.
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100(mod5) simplify it
Answer:
0
Step-by-step explanation:
|2x+2| =< 0
If the solution set is”null” also known as the empty set or no solution. Explain why
=========================================================
Explanation:
The output of an absolute value is never negative.
The result of |2x+2| can never be negative, so we will have no solutions for the portion |2x+2| < 0
There will be one solution for |2x+2| = 0 as shown in the steps below
|2x+2| = 0
2x+2 = 0
2x = -2
x = -2/2
x = -1
------------------
Check:
|2x+2| ≤ 0
|2(-1)+2| ≤ 0
|-2+2| ≤ 0
|0| ≤ 0
0 ≤ 0
We get a true statement at the end to fully confirm the single solution of x = -1
PLEASE I ONLY HAVE 15 MINS LEFT, HELP! Find the inverse of the following matrix:
Answer:
The inverse is [-4/3 -5/3 1]
[-4/3 -8/3 1]
[1/3 2/3 0]
Sorry, I didn't know how to type it well.
Find the product of 2√/18 and 2√5 in simplest form. Also, determine whether the
result is rational or irrational and explain your answer.
2√10/3 is irrational number
The term "rational number" refers to any number that can be stated as a fraction, as well as a positive, negative, or zero. If q is not equal to zero, it can be expressed as p/q.
The word "rational" is derived from the word "ratio," which is a fraction and really refers to a comparison of two or more values or integer numbers. It is the ratio of two integers, to put it simply.
Consider the rational integer 3/2. It denotes that the integer 3 is divided by the number 2.
Irrational numbers are those that are not rational in nature. Let's explain; irrational numbers can be expressed as decimals but not as fractions, hence they cannot be expressed as the ratio of two integers.
After the decimal point, irrational numbers have an infinite amount of non-repeating digits. Here's an illustration of an irrational number:
For instance, 8 = 2.828
MULTIPLYING THE TWO NUMBERS,
2/√18 x 2√5
= 2 /3√2 x 2√5
= 4√5/3√2
= 2√2 x √5 /3
= 2√10/3
rational number multiplied by an irrational number is always a irrational number.
Since √10 is a irrational number, therefore, 2√10/3 is irrational number.
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