Find the general solution of the following differential equation. primes denote derivatives with respect to x.3x^2y' +6xy = 15y^3

For  y=sin [(x – 1)^2(x + 1)], the differentiation of y will be

y'=cos[(x-1)^2(x+1)]((x-1)^2+2(x^2-1)).

What is differentiation?

In calculus, differentiation is one of the two important concepts apart from integration. Differentiation is a method of finding the derivative of a function. Differentiation is a process, in Maths, where we find the instantaneous rate of change in function based on one of its variables. The most common example is the rate change of displacement with respect to time, called velocity. The opposite of finding a derivative is anti-differentiation.

If x is a variable and y is another variable, then the rate of change of x with respect to y is given by dy/dx. This is the general expression of derivative of a function and is represented as f'(x) = dy/dx, where y = f(x) is any function.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

dy/dx

Now,

Given function

Y=sin[(x-1)^2(x+1)]

therefore its differentiation is

y'=cos[(x-1)^2(x+1)]((x-1)^2*1+(x+1)*2*(x-1))

y'=cos[(x-1)^2(x+1)]((x-1)^2+2(x^2-1))

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Right Question:-

Which of the following expressions is y' given that y=sin [(x – 1)^2(x + 1)]

The correct answer is E: y' = (3x – 1) (2π – 1) cos [(x – 1)π (x + 1)].

The derivative of a function is a mathematical concept that describes the rate of change of the function at a particular point. It represents the slope of the function at that point and provides information about how the function is changing with respect to its input.

Formally, the derivative of a function f(x) at a point x is defined as the limit of the change in the function value as the change in the input approaches zero, i.e.:

f'(x) = lim h -> 0 (f(x + h) - f(x)) / h

The derivative can be calculated using various techniques, including differential calculus and numerical methods.

The derivative has many applications in physics, engineering, and economics, among other fields. For example, in physics, the derivative of the position function of an object gives its velocity, while the derivative of the velocity function gives its acceleration. In finance, the derivative of a stock price function gives the rate of change of the stock price, which is an important measure of risk.

The derivative of y = sin [(x – 1)π (x + 1)], using the chain rule, is:

y' = cos [(x – 1)π (x + 1)] * [(x + 1)π – (x – 1)π]

= cos [(x – 1)π (x + 1)] * 2π

= 2π cos [(x – 1)π (x + 1)]

So, the correct answer is E: y' = (3x – 1) (2π – 1) cos [(x – 1)π (x + 1)].

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a) There are 10 different combinations of pairs of coins.

b) List of the combinations:

(10p, 1p), (10p, 50p), (10p, 2p), (10p, 20p), (1p, 50p), (1p, 2p), (1p, 20p), (50p, 2p), (50p, 20p), (2p, 20p)

A combination is a selection of items from a set that has distinct members, such that the order of selection does not matter.

Given,

Angela has coins of

10p, 1p, 50p, 2p, 20p

a) Combination of pulling out two coins

= ⁵C₂

= (5!)/(5-2)!×(2!)

= 5 × 4 × 3 × 2 × 1/3 × 2 × 1 × 2 × 1

= 120/12

= 10

b) Taking 10p as one coin

(10p, 1p), (10p, 50p), (10p, 2p), (10p, 20p),

Taking 1p as one coin, as repetition is not allowed

(1p, 50p), (1p, 2p), (1p, 20p),

Taking 50p as one coin, as repetition is not allowed

(50p, 2p), (50p, 20p),

Taking 2p as one coin, as repetition is not allowed

(2p, 20p)

a) Hence, 10 is the total number of combinations of pairs of coins

b) All combinations are:

(10p, 1p), (10p, 50p), (10p, 2p), (10p, 20p), (1p, 50p), (1p, 2p), (1p, 20p), (50p, 2p), (50p, 20p), (2p, 20p)

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The complete question is as follows:

Angela has the following coins in her pocket:

10p 1p 50p 2p 20p

She pulls out two coins at random.

a) How many different combinations of pairs of coins are there

b) List all of the combinations,

write your answers like this: (10p, 1p)

You must write all of your answers on one line.

The first four terms of the Taylor series for the right-hand side of the Gompertz equation, f(y) = ry ln(ky), about y = k  is: f(y) = rk ln(kk) + (y - k)(2r) + (y - k)² / 2! (r / k) + (y - k)³ / 3! (-r / k²).

To determine this :

The right-hand side of the Gompertz equation is f(y) = ry ln(ky). To find the first four terms of the Taylor series for this f(y) about y = k, we can use the formula:

f(y) = f(k) + (y - k)f'(k) + (y - k)² / 2! f''(k) + (y - k)³ / 3! f'''(k) + ...

Where f'(k), f''(k), and f'''(k) are the first, second, and third derivatives of f(y) evaluated at y = k, respectively.

First, we need to find the derivatives of f(y):

f'(y) = r ln(ky) + r

f''(y) = r / y

f'''(y) = -r / y²

Next, we evaluate these derivatives at y = k:

f'(k) = r ln(kk) + r = 2r

f''(k) = r / k

f'''(k) = -r / k²

Finally, we substitute these values into the Taylor series formula:

f(y) = f(k) + (y - k)f'(k) + (y - k)² / 2! f''(k) + (y - k)³ / 3! f'''(k)

f(y) = rk ln(kk) + (y - k)(2r) + (y - k)² / 2! (r / k) + (y - k)³ / 3! (-r / k²)

This is the first four terms of the Taylor series for f(y) about y = k.

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The square root of a negative number is not a real number, the equation has no real solution.

The equation −4 = √(2-x)−2 has no real solution because the square root of a negative number is not a real number.

If we expand the equation, we get:

-4 = √(2-x) - 2

-4 + 2 = √(2-x)

-6 = √(2-x)

Squaring both sides of the equation, we get:

36 = 2-x

x = -34

Since the square root of a negative number is not a real number, the equation has no real solution.

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The value of tan C is 5/12.

There are three commonly used trigonometric identities.

Sin x = 1/ cosec x

Cos x = 1/ sec x

Tan x = 1/ cot x or sin x / cos x

Cot x = cos x / sin x

We have,

From the figure,

BA = 15

BC = 36

AC = 39

So,

Tan C = BA / BC = 15/36 = 5/12

Thus,

Tan C is 5/12.

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Answer:

0.41 cents if youre adding

Step-by-step explanation:

Based on the information, we can infer that the area of this trapezoid is 192 in² (option D).

To calculate the area of the trapezoid it is necessary to know the length of the bases and the height. Once we know these data we can complete the following formula to find the area:

A = 1/2 (b1 + b2) h

b1 = length of the lower base

b2 = length of the upper base

h = height

So, we just have to plug in the values to find the area of this trapezoid.

A = 1/2 (18in + 14in) 12in

A = 192in²

Based on the above, the area of this trapezoid would be 192 in² (option D).

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Answer:

120

Step-by-step explanation: 125.4/104.5x100

Therefore , the solution of the given problem of velocity comes out to be  33.94 ft/sec is the velocity.

Looking at something from a distance allows us to calculate its speed. An object's speed determines how far it can travel in a certain amount of time. Utilizing the equation velocity = distance/time, velocity is computed. Miles per hour (mph), kilometers an hour (km/h), and meters a second (m/s) are the three most common units of speed (mph).

Here,

Given :

=>  v = (8h) ^ 1/2

=>  v = √8h

We separate the variables whose values we want to be determined in the aforementioned problems.

(1) We change the variable to become

  v = √8h  in order to solve for v.

Using the supplied as a replacement,

v = √8 * 144 = 33.94 ft/sec

Therefore , the solution of the given problem of velocity comes out to be  33.94 ft/sec is the velocity.

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The amount of fertilizer Debra needs is 14 cups.

Area of a two dimensional shape is the total region which is bounded by the object's shape.

Given that,

Debra's rectangular vegetable garden measures 9 1/3 yard by 12 yard.

Area of a rectangle = Length × Width

Area of the rectangular garden = 12 × 9 1/3

                                                    = 12 × 28/3

                                                    = 112 square yard

Number of cups of fertilizer needed for 1 square yard = 1/8 cups

Number of cups of fertilizer needed for 112 square yard = (1/8) × 112

                                                                                             = 14 cups

Hence 14 cups of fertilizer are needed for the whole garden.

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The frequency distribution of the digits indicates that the heights were reported rather than actually measured. This is because the frequency of the digits is not evenly distributed, which would be expected for a set of actual measurements.

Frequency Distribution:

Digit Frequency

0 3

1 5

2 9

3 5

4 3

5 4

6 4

7 4

8 2

9 1

From the distribution of the digits, it appears that the heights were reported rather than actually measured. This is because the frequency of the digits is not evenly distributed, which would be expected for a set of actual measurements.

The frequency distribution of the digits indicates that the heights were reported rather than actually measured. This is because the frequency of the digits is not evenly distributed, which would be expected for a set of actual measurements. For example, the frequency of the digit '2' is more than twice as high as the frequency of the digits '1' and '3', which is not likely to occur in a set of actual measurements.

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Total 39 newspapers must be collected to be certain that at least two newspapers have the same number of pages.

A typical newspaper has 9 pages minimum and 46 pages maximum.

We must determine the maximum number of newspapers that can be published with various page counts

9, 10, 11, 12 .......................... 45, 46

All numbers are in arithmetic order, as we can see

The first term of series(a(1)) = 9

The difference between terms = a(2) - a(1)

The difference between terms = 10 - 9

The difference between terms = 1

The formula of nth term of arithmetic series

a(n) = a + (n - 1)d

we know a(n) = 46

Now putting the values

46 = 9 + (n - 1)1

Now solve

46 = 9 + n - 1

46 = 8 + n

Subtract 8 on both side, we get

n = 38

Then, newspapers must be collected to be certain that at least two newspapers have the same number of pages = 38 + 1 = 39

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Using row elimination method the solution of the system equation is [tex]x_1=12,x_2=-7[/tex].

What is equation?

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

Here the given system equation is

[tex]2x_1+4x_2=-4[/tex]

[tex]5x_1+7x_2=11[/tex]

The augmented matrix of the given equation is

=> B = [A ,b] = [tex]\left[\begin{array}{ccc}2&4&-4\\5&7&11\\\end{array}\right][/tex]

Multiply the 1st row by 1/2 then

=> [tex]\left[\begin{array}{ccc}1&2&-2\\5&7&11\\\end{array}\right][/tex]

Add -5times the first row to the second row then

=> [tex]\left[\begin{array}{ccc}1&2&-2\\0&-3&21\\\end{array}\right][/tex]

Multiply 2nd row by -1/3 then

=> [tex]\left[\begin{array}{ccc}1&2&-2\\0&1&-7\\\end{array}\right][/tex]

Add 2 times the second row to the first row then

=> [tex]\left[\begin{array}{ccc}1&0&12\\0&1&-7\\\end{array}\right][/tex]

Hence the given system equation reduce to [tex]x_1=12 , x_2=-7[/tex] which is required solution.

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Answer:

Step-by-step explanation:

The slope of a line perpendicular to another line with slope m is the negative reciprocal of m. So if the slope of line G is 5/4, the slope of line H, which is perpendicular to line G, would be -4/5.

Cardi B paid back the loan on February 3rd + 122.72 days = May 16th.

Interest is a fee charged by a lender to a borrower for the use of money. It represents the cost of borrowing money and is typically expressed as a percentage of the loan amount. Interest is paid over a period of time and is calculated based on the principal amount of the loan, the interest rate, and the length of the loan term.

Simple interest is calculated only on the principal amount of the loan, while compound interest is calculated on both the principal and the accumulated interest over time. Compound interest results in a faster growth of the loan balance compared to simple interest.

Let's call the number of days Cardi B had the loan "d". The exact interest can be calculated as follows:

Interest = Principal * Rate * (Number of days / 365)

From the information provided, we know that the maturity value was $17,873.97 and the principal was $17,500.00, so the interest can be calculated as follows:

Interest = $17,873.97 - $17,500.00 = $373.97

We can now use the formula for exact interest to find the number of days:

$373.97 = $17,500.00 * 0.065 * (d / 365)

Solving for d:

d = 365 * ($373.97 / ($17,500.00 * 0.065))

d = 365 * ($373.97 / $1137.50)

d = 365 * (0.3288)

d = 122.72

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An equation y = a(x - h)² + k is equivalent to the formula a = (y - k)/(x - h)², So Option D is correct

An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.

For example, 3x+2y=0.

Types of equation

1. Linear Equation

2. Quadratic Equation

3. Cubic Equation

Given that,

The equation,

y = a(x - h)² + k

Equivalent equation  = ?

by using rules of algebra, we can transform given equation,

the rules of algebra

1. If a number is in division operation then when it goes to another side of equal to, it will be in multiplication operation or vice-versa.

2. If a number is in addition  operation then when it goes to another side of equal to, it will be in subtraction operation or vice-versa.

So,

by transferring k to the left hand side,

y - k = a(x - h)²

Now, by transferring (x - h)² to the left hand side,

(y - k)/(x - h)² =  a

Hence, the equivalent equation is, a = (y - k)/(x - h)²

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The equation of the sphere with the given endpoints is: (x - x0)² + (y - y0)² + (z - z0)² = r²

where x0 = (x1 + x2) / 2, y0 = (y1 + y2) / 2, z0 = (z1 + z2) / 2, and r = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²) / 2.

All points on a sphere are at a set distance from the centre, making it a three-dimensional geometric object. It is thought to be one of the most basic three-dimensional shapes and is a perfect circle.

A sphere with centre (x, y, z) and radius r has the equation:

(x - x0)² + (y - y0)² + (z - z0)² = r²

If one of the diameters of the sphere has endpoints (x1, y1, z1) and (x2, y2, z2), then the center of the sphere can be found by finding the midpoint of the two endpoints:

x0 = (x1 + x2) / 2

y0 = (y1 + y2) / 2

z0 = (z1 + z2) / 2

The radius can be found by finding the distance between the two endpoints:

r = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²) / 2

Therefore, the equation of the sphere with the given endpoints is:

(x - x0)² + (y - y0)² + (z - z0)² = r²

where x0 = (x1 + x2) / 2, y0 = (y1 + y2) / 2, z0 = (z1 + z2) / 2, and r = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²) / 2.

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A point in the solution of the system of inequalities is (2, 0.5)

From the question, we have the following parameters that can be used in our computation:

21x>5y+49x>2y>2-28x>2y

Express properly

So, we have the following representation

21x > 5y+49

x > 2y

2 - 28x > 2y

Next, we plot the graphs of the inequality expressions on a plane

(See attachment for the graph)

The shaded region represent the solution

And a point in this region is (2, 0.5)

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Answer:

Equally likely and Unlikely, I would say.

Step-by-step explanation:

Think about it. You have a 50/50 chance of getting a terrier. 50 and 50 are equal. Likely and Unlikely. I'd say that's the answer and that's my reasoning.

Answer:

                    likely

the answer is ^

a) The like terms are B, D, F

b) The equivalent expression is option A

c) The little 5 outside would multiply all the exponents. Option D

We know that the like terms are the terms that have the same kind of algebraic exponents and can easily be added or subtracted as the case may be. Hence we have to look out for the factors that do have the same exponents.

We would have in the second question;

3x^2 + 2x - 4 + 5x^2 - 4x + 5

Collecting the like terms;

3x^2  + 5x^2 +  2x -  4x - 4 + 5

8x^2 - 2x + 1

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The number of shirts at which the cost would be same is 12. As, $352 < $380.75, First Company offers a better deal for an order of 35 shirts.

Charge of first company is $9.65 per shirt plus setup fee of $43 &

Charge of another company is $8.40 per shirt plus setup fee of $58

The number of shirts such that the cost of both the companies are same.

Let the required number of shirts be n which can be evaluated as following:

Cost of first company = Cost of another company

⇒ 9.65n + 43 = 8.40n + 58

⇒ 9.65n−8.40n = 58−43

⇒ 1.25n = 15

⇒ n = 151.25

⇒ n = 12

Thus, the number of shirts at which the cost would be same is 12.

First company charges $9.65 per shirt + setup fee of $43

35 shirts = 35 × 9.65 + 43

              = $380.75

Second company charges $8.40 per shirt + setup fee of $58

35 shirts = 35 × 8.40 + 58

              = $352

As, $352 < $380.75, First Company offers a better deal for an order of 35 shirts.

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The height of the building is 120 ft.

According to Pythagoras theorem, we can write the relation between hypotenuse, base and perpendicular as -

(hypotenuse)² = (base)² + (perpendicular)²

(h)² = (b)² + (p)²                                                        ... Eq { 1 }

Given is that Barrett is looking at a building on the school grounds which is shaped like a rectangle as shown. He knows the building has a height of {25x} feet and a width of {28x} feet with a diagonal of 180 feet.

We can apply the Pythagoras theorem to the building as -

(diagonal)² = (height)² + (width)²

180 x 180 = 625x² + 784x²

32400 = 1409x²

x² = 32400/1409

x² = 23

x = 4.8                                   ... Eq { 1 }

Height of the building would be -

{h} = 25x = 25 x 4.8

{h} = 120 ft

Therefore, the height of the building is 120 ft.

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The positive difference between the two probabilities is 11/35.

To calculate the probability that three distinct numbers chosen at random without replacement from the set would have an even sum, Jan would consider the number of ways to choose three even numbers (4 choose 3) and divide that by the total number of combinations of three numbers (7 choose 3).

To calculate the probability that three distinct numbers chosen at random without replacement from the set would have an even product, Judy would consider the number of ways to choose three numbers such that the product is even (3 choose 2) and divide that by the total number of combinations of three numbers (7 choose 3).

The positive difference between the two probabilities would be the difference between these two calculations. In this case, 11/35 is a positive difference, meaning that the probability of having an even product is 11/35 higher than the probability of having an even sum.

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The area of the phrase 1 3/4 X 2 1/2 when using an area model is 4 3/8.

A method of dividing an area into smaller portions is the area model.The example provided below,

Distance = 27 = 20 + 7

length = 63 = 60 + 3

Area = (20 + 7) x (60 + 3)

= 1200 + 60 + 420 +21

= 1701 unit²

You can solve using an area model.

Ours was given

It follows that the area of the expression 1 3/4 X 2 1/2 using an area model is 4 3/8.

The complete question is,

You can solve using an area model. 1 3/4 X 2 1/2

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Velocity is the measure of the rate of change of position, or the speed of an object in a specific direction.

Since you drove from home to school and then back home, the velocity on the return home would be different than the velocity on the way to school, since the direction of travel is different.

The average speed for the trip can be calculated by dividing the total distance (25 miles) by the total time (1 hour). Therefore, the average speed of the trip is 25 miles per hour.

Acceleration is the rate of change of velocity, or the rate at which an object speeds up or slows down. In this case, you would have to accelerate when you reach the school in order to pick up your child and then accelerate again when you leave the school in order to return home.

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Answer: Yes as long as one of the two mixed numbers is negative

Step-by-step explanation:

Answer: Yes, the sum of two mixed numbers can be equal to 2 but only of one of the mixed numbers is negative

Step-by-step explanation:

Suri is right. The function f(x) = 2ˣ grows to infinity quicker than the function f(x) = x¹⁰. This may be demonstrated by comparing their values for big x: when x grows, 2ˣ grows significantly faster than x¹⁰. This suggests that 2ˣ approaches infinity more quickly than x¹⁰.

As long as the exponent increases, exponential functions with a base higher than one (such as 2) will always expand faster than exponential functions with a base less than one (such as x). Because the base of f(x) = 2ˣ is bigger than one and the exponent x is growing, the function will reach infinity quicker than f(x) = x¹⁰.

Here,

Suri is correct. The function f(x) = 2ˣ increases to infinity faster than f(x) = x¹⁰. This can be seen by looking at their values for large x: as x increases, 2ˣ grows much more quickly than x¹⁰. This means that 2ˣ approaches infinity faster than x¹⁰.

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Answer:

1. Order from least to greatest:  1/4 cup sugar, 1/2 cup walnuts, 2/3 cup strawberries, 3/4 cup of flour

2. Sugar is the ingredient the recipe requires the least amount

Step-by-step explanation:

1.

2/3 cup of strawberries,  1/4 cup sugar, 1/2 cup walnuts, and 3/4 cup of

flour.

8/12 cup of strawberries, 3/12 cup sugar, 6/12 cup walnuts, and 9/12 cup of flour

Order from least to greatest:  1/4 cup sugar, 1/2 cup walnuts, 2/3 cup strawberries, 3/4 cup of flour

2.

Sugar is the ingredient the recipe require the least amount

Answer:

Step-by-step explanation:

Pi (π) is an irrational number that represents the ratio of the circumference of a circle to its diameter. Pi cannot be expressed as a simple fraction, so it's an irrational number. Pi is approximately equal to 3.14159...

The square root of 2 is another example of an irrational number. This number cannot be expressed as a simple fraction either, so it's also irrational. The square root of 2 is approximately equal to 1.41.

The number e is greater than 5 and is also an irrational number. e represents the base of the natural logarithm and is approximately equal to 2.71828...