Find the general solution to the differential equation y′ = x − x sin²
Answers
The general solution to the differential equation y' = x - x × sin²(x) is y = x + C, where C is a constant.
To find the general solution to the given differential equation, we'll separate the variables and integrate both sides.
The differential equation is: y' = x - x×sin²(x)
Step 1: Separate the variables
We can rewrite the equation as:
dy = (x - x×sin²(x)) dx
Step 2: Integrate both sides
Integrating the left side with respect to y gives us just y:
∫dy = ∫dx
On the right side, we need to integrate the expression (x - x×sin²(x)) with respect to x. This requires a bit more work.
Step 3: Expand the integrand
x - xsin²(x) can be expanded as follows:
x - xsin²(x) = x - x×(1 - cos²(x))
= x - x + xcos²(x)
= xcos²(x)
Step 4: Integrate xcos²(x) with respect to x
To integrate xcos²(x), we'll use integration by parts. Let's choose u = x and dv = cos²(x) dx.
Differentiating u, we get du = dx, and integrating dv, we have:
∫cos²(x) dx = ∫dv = v = (1/2)(x + sin(2x)/2)
Using the formula for integration by parts:
∫u dv = uv - ∫v du
We have:
∫x×cos²(x) dx = (1/2)(x + sin(2x)/2) - ∫(1/2)(x + sin(2x)/2) dx
Simplifying:
∫x×cos²(x) dx = (1/2)(x + sin(2x)/2) - (1/2)∫(x + sin(2x)/2) dx
We can integrate the remaining term on the right side.
Step 5: Integrate (x + sin(2x)/2) with respect to x
∫(x + sin(2x)/2) dx = (1/2)x² + (1/4)sin(2x) + C
Where C is the constant of integration.
Now, let's substitute this result back into our original equation:
y = ∫dx + C
= x + C
The general solution to the given differential equation is y = x + C, where C is a constant.
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Related Questions
pls help i will rate u 5 stars and possibly mark u brainless HELP PLS.
Answers
Is this relation a function? Justify your answer. A. Yes, because the number of x-values is the same as the number as the y-values B. No, because two point with the same y-value have different x-values.
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Hope this will help.
URGENT!! ILL GIVE
BRAINLIEST! AND 100 POINTS
Irena Surveyed 100 people. She asked each person two questions
Answers
Answer: b
Step-by-step explanation: trust me i have done that question before
Find the sum of the series
1 - 2/3 + 4/9 - 8/27+16/81 - 32/243 + ....
Answers
The sum of the series:
1 - 2/3 + 4/9 - 8/27+16/81 - 32/243 + .... = 2/3
All geometric sequences have a starting term a and a common ratio r.
The common ratio r is the number that the terms of the sequence are multiplied to find the next term, hence the name "ratio". All terms in the sequence are the same, hence the name "common".
The common ratios here are the numbers multiplied by 2/3 to get −4/9
and the numbers multiplied by−4/9 to get 8/27.
2/3⋅ 2/3 = \(\frac{2*2}{3*3 }\) = 4/9 .
2/3 × (-2/3) = -4/9
The infinite sum of the geometric series can only be found in a specific case: r lies between −1 and 1.
When r is between these values, adding the series will converge - close on both sides - to a certain number.
Otherwise the sum diverges - further apart - i.e. there is no specific value that the series approaches. is in This means that an infinite series can be computed as the number to which the series converges.
This calculation is done using the formula
S∞ = a₁ − r.
a, the first term of the sequence is 2/3.
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Eliza fell asleep at 9:35PM. She awoke 8 hours 45 minutes later. What time did Eliza wake up?
Answers
Answer:
She would have woken up at 6:20 AM
Step-by-step explanation:
Answer:
6:20am
Step-by-step explanation:
Hope this helps!
What is the answer to x? Given the top of the triangle is 80.
Answers
After creating the balance sheet, Roberto decided to use his investments to pay off his car loan. How will that decision affect the difference between his assets and liabilities? It will make the assets $5,000 less than the liabilities. It will make the assets $5,000 more than the liabilities. The difference between the assets and the liabilities will remain the same. The difference between the assets and the liabilities cannot be compared.
Answers
Answer: D
Step-by-step explanation:
Answer:
Step-by-step explanation:
The difference between the assets and the liabilities cannot be compared.
Use synthetic division to find the result when x³ + 7x² - 12x + 14 is divided by a 1. If there is a remainder, express the result in the form q(x) + b(x)*
Answers
Using synthetic division, we can divide the polynomial x³ + 7x² - 12x + 14 by the divisor 1. Performing the synthetic division, we obtain a quotient of x² + 6x - 6 and a remainder of 8.
To divide the polynomial x³ + 7x² - 12x + 14 by the divisor 1 using synthetic division, we set up the synthetic division table as follows:
1 | 1 7 -12 14
We begin by bringing down the coefficient of the first term, which is 1, and place it on the line below the division bar. Then we multiply the divisor, 1, by the value we brought down and write the result under the next coefficient. Adding the values in the second row, we obtain the new value. We continue this process for each term until we reach the last term.
1 | 1 7 -12 14
1 8 -4 10
The values in the last row represent the coefficients of the quotient polynomial. Therefore, the quotient is x² + 6x - 6. The remainder, which is the last value in the last row, is 10. Since the divisor is 1, the remainder does not affect the quotient. Hence, the result of the division is x² + 6x - 6 with a remainder of 10.
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whats the answer to 4a-3b also a=7 and b=5
Answers
Answer:
The answer is 13
Step-by-step explanation:
4(7)-3(5)
4(7)=28
28-3(5)
3(5)=15
28-15= 13
Hope this helped!
Answer:
Step-by-step explanation:
7 x 4 = 28
3 x 5 = 15
28 - 15 = 13
I hope this helps
I'm not very good at explaining
ayuda!!
lo necesito antes del 17 de febrero jeje
Answers
50/100 = 1/2
porfa dame brilliant answer
A researcher wanted to estimate the mean number of hours adults spend formally exercising each week. She gathered a random sample and created a 95% confidence interval of (0.45 hours, 7.94 hours). Which of the following is the correct interpretation of this confidence interval? Select one: a. We are 95% confident that the population mean number of hours adults spend on formal exercise each week lies between 0.45 and 7.94. b. There is a 0.95 probability that adults exercise formally between 0.45 hours and 7.94 hours per week. c. The sample mean number of hours adults spend on formal exercise each week lies between 0.45 and 7.94. d. The population mean number of hours adults spend on formal exercise each week lies between 0.45 and 7.94. e. We are 95% confident that the sample mean number of hours adults spend on formal exercise each week lies between 0.45 and 7.94.
Answers
The correct interpretation of the confidence interval is: “We are 95% confident that the population mean number of hours adults spend on formal exercise each week lies between 0.45 and 7.94.”
Option (a) is the correct interpretation because a confidence interval provides a range of values within which the true population mean is likely to fall. In this case, based on the researcher’s sample and statistical analysis, there is a 95% confidence that the true population mean number of hours adults spend on formal exercise per week is between 0.45 and 7.94 hours.
This interpretation takes into account the uncertainty inherent in statistical estimation and provides a range rather than a specific value. The other options either refer to the sample mean or imply probabilities, which are not accurate interpretations of a confidence interval.
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if a fair die is rolled 5 times, what is the probability, rounded to the nearest thousandth, of getting at least 2 fours?
Answers
So the probability, rounded to the nearest thousandth, of getting at least 2 fours in 5 rolls of a fair die is 0.194.
What is the simple definition of probability?A probability is a number that expresses the possibility or likelihood that a specific event will take place. Probabilities can be stated as proportions with a range of 0 to 1, or as percentages with a range of 0% to 100%.
According to the given information:The probability of getting at least 2 fours is the sum of the probabilities of getting exactly 2, 3, 4, or 5 fours:
P(X ≥ 2) = P(X=2) + P(X=3) + P(X=4) + P(X=5)
Using the binomial formula, we can calculate each of these probabilities:
P(X=k) = (n choose k) p^k (1-p)^(n-k)
where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from n distinct items.
P(X=2) = (5 choose 2) (1/6)² (5/6)³ = 0.1608
P(X=3) = (5 choose 3) (1/6)³ (5/6)² = 0.0322
P(X=4) = (5 choose 4) (1/6)⁴ (5/6)¹ = 0.0013
P(X=5) = (5 choose 5) (1/6)⁵ (5/6)⁰ = 0.00003
Therefore,
P(X ≥ 2) = 0.1608 + 0.0322 + 0.0013 + 0.00003 = 0.1943
So the probability, rounded to the nearest thousandth, of getting at least 2 fours in 5 rolls of a fair die is 0.194.
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That is, what qualities does a taylor polynomial havethat the tangent line does not necessarily have?
Answers
The quality which a taylor polynomial have that the tangent line does not necessarily have is that Taylor polynomial has a far better approximation of f(x) near x = a than is the tangent line
What is Taylor polynomial ?Taylor polynomial function simply refers to an infinite sum of terms that are expressed in terms of the function's derivatives at a single point.
So therefore, the quality which a taylor polynomial have that the tangent line does not necessarily have is that Taylor polynomial has a far better approximation of f(x) near x = a than is the tangent line
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Which of the following is a quantitative variable?
a) the texture of a rug
b) the breed of a dog
c) the numbers of toppings on a pizza
d) the colour of a person’s hair
Answers
C is the most accurate one i guess
Find the coordinates of the point 7/10 of the way from A to B
A(-4,-5) B(12,4)
Answers
Answer:
the coordinates of the point 7/10 of the way from A to B are (7.2 , 1.3)
Step-by-step explanation:
Given:
A(-4,-5)
B(12,4)
To find:
Find the coordinates of the point 7/10 of the way from A to B
Solution:
Let x and y be the coordinate of the point 7/10 of way from A to B
x = -4 + (7/10) (12 - (-4)
= -4 + 7/10 (12+4)
= -4 + 7/10 (16)
= -4 + 112/10
= -4 + 11.2
x = 7.2
y = -5 + (7/10) (4-(-5))
= -5 + (7/10) (4+5)
= -5 + (7/10) (9)
= -5 + 63/10
= -5 + 6.3
y = 1.3
One way to find the coordinates of the point 7/10 of the way from A to B is:
A + (7/10) (B - A) = (7/10)B + (3/10)A
(7B + 3A)/10 = (7(12,4) + 3(-4,-5)) / 10
= (84, 28 + -12, -15) / 10
= (84 -12, 28 -15) /10
= (72 , 13) / 10
= (72/10 , 13/10)
= (7.2 , 1.3)
Hence the coordinates are (7.2,1.3)
Complete the table for y=1/5x+1 and graph the resulting line
Answers
Answer:
yidyf
Step-by-step explanation:
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can anyone help me with these questions
Answers
Answer:
Nah
Step-by-step explanation:
DUE TODAY PLEASE HELP WELL WRITTEN ANSWERS ONLY!!!!
Answers
Answer:
\(r = \sqrt{ {(6 - 2)}^{2} + {(1 - ( - 3))}^{2} } \)
\(r = \sqrt{ {4}^{2} + {4}^{2} } = \sqrt{16 + 16} = \sqrt{32} \)
So the equation of the circle is
\( {(x - 2)}^{2} + {(y + 3)}^{2} = 32\)
Are solar systems unique to just our sun? Explain.
Answers
Can someone help on this question from khanademy geometry. Finding the equation for the circle
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The equation of the circle graphed in this problem is:
(x - 5)² + y² = 16.
What is the equation of a circle?The equation of a circle of center \((x_0, y_0)\) and radius r is given by:
\((x - x_0)^2 + (y - y_0)^2 = r^2\)
For the circle graphed, we have that:
The center is at point (5,0), hence \(x_0 = 5, y_0 = 0\).The radius is of 4 units.Hence the equation of the circle graphed is:
(x - 5)² + y² = 16.
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which point has coordinates
Answers
Answer:
The coordinates of a point are a pair of numbers that define its exact location on a two-dimensional plane. Recall that the coordinate plane has two axes at right angles to each other, called the x and y axis.
Teddy is preparing to sell some wooden blocks at a garage sale. Before pricing individual bags of blocks, he decides to determine the approximate number of blocks per pound by researching the weights of different sets of blocks. The graph shows his data.
A graph shows weight of set (pounds) labeled 0.5 to 5 on the horizontal axis and number of blocks on the vertical axis. A line increases from 0 to 5.5.
About how many blocks are there per pound? Round to the nearest block.
10
20
30
40
Answers
Answer:
B. 20
Step-by-step explanation:
took the quiz
The approximate number of blocks per pound is 20, the correct option is B.
What is the unitary method?The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value. Unitary method is a technique by which we find the value of a single unit from the value of multiple devices and the value of more than one unit from the value of a single unit. It is a method that we use for most of the calculations in math.
We are given that;
The graph labeled 0.5 to 5 on the horizontal axis
Now,
To find the slope, we can use two points on the line and apply the formula:
slope = (change in number of blocks) / (change in weight)
One point we can use is (0.5, 10), which means that a set of 10 blocks weighs 0.5 pounds. Another point we can use is (2.5, 50), which means that a set of 50 blocks weighs 2.5 pounds.
Using these points, we get:
slope = (50 - 10) / (2.5 - 0.5)
slope = 40 / 2
slope = 20
Therefore, by unitary method the answer will be 20.
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Find the measure of the arc.
A
146°
E
B
MABC = [ ? 1°
Answers
Answer:
i get mABC = 270°
Step-by-step explanation:
This is fairly straight forward once you the right angle symbol for the arc AC. Also realizing the 148° for mAB is not needed for this question. The 148° for mAB would be useful if you wanted to calculate arc mBC.
Remember there are 360° in a circle.
The arc mAC is 90°
The arc mABC is the rest of the circle. is the arc from A to B to C
arc ABC + arc AC is the full circle 360°
mABC + mAC = 360° mAC = 90°
mABC + 90° = 360° - 90° from both sides
mABC = 360° - 90°
mABC = 270°
The angle ∠ABC in the given circle will be 270°.
What is an angle?
An angle is a geometry in plane geometry that is created by 2 rays or lines that have an identical terminus.
Angle is very useful to measure any angular rotational motion for example if you have to find out the rotation among the circle then it cannot measure in length parameter such that meter inch so it will be measured as an angle in a unit of degree or radian.
In another word, the angle is the measurement of the angular distance for example for linear motion we have a meter inch but for angular rotation, we don't have the measurement so the angle is useful to measure the angular rotation.
Now since we know that the angle of complete rotation is 360°.
Now ∠ADC + ∠ABC = 360
Given that ∠ADC = 90°
∠ABC = 360° - 90° = 270°
Hence, 270° will be the correct answer.
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find the total surface area, leave your terms in pi.
Answers
Answer:
75.39822
Step-by-step explanation:
A rectangular slab on grade is 60 ft 0 in. long × 45 ft 0 in. wide. What is the diagonal measurement in feet and inches?
A. 52 ft 6 in.
B. 75 ft 0 in.
C. 105 ft 8 in.
D. 115 ft 11 in.
Answers
The diagonal measurement as √5625 ft, which is approximately 75 feet, the correct answer is B. 75 ft 0 in.
The diagonal measurement of the rectangular slab on grade can be found using the Pythagorean theorem. The diagonal is the hypotenuse of a right triangle formed by the length and width of the slab.
To calculate the diagonal measurement, we can apply the Pythagorean theorem:
Diagonal² = Length² + Width²
Substituting the given values, we have:
Diagonal² = (60 ft 0 in.)² + (45 ft 0 in.)²
Calculating this expression, we find:
Diagonal² = 3600 ft² + 2025 ft²
Diagonal² = 5625 ft²
Taking the square root of both sides, we obtain:
Diagonal = √5625 ft
Diagonal ≈ 75 ft
Therefore, the diagonal measurement of the rectangular slab on grade is approximately 75 feet.
To find the diagonal measurement of the rectangular slab on grade, we can use the Pythagorean theorem,
which states that in a right triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (length and width).
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What changes when the parent function \(f(x)=x^2\) is stretched by a positive factor?
Answers
9514 1404 393
Answer:
vertical stretch: the function value (f) is multiplied by the stretch factorhorizontal stretch: the variable (x) is divided by the stretch factorStep-by-step explanation:
A stretch can be vertical, horizontal, or both.
Multiplying the function value by a factor greater than 1 will stretch the graph vertically.
Dividing the variable value by a factor greater than 1 will stretch the graph horizontally.
__
The graph is stretched by the factor according to how it is applied.
![What changes when the parent function [tex]f(x)=x^2[/tex] is stretched by a positive factor?](https://i5t5.c14.e2-1.dev/h-images-qa/answers/attachments/bUcFvAkzUKakieNmyNwUXoZVfdgDKQYq.png)
2. What is the slope of a line perpendicular to the line y=-4x + 16?
1/4
-4
4
-1/4
Answers
The slope of a line perpendicular to y=-4x + 16 is (1/4), so the correct option is the first one (counting from the top).
How to get the slope of the perpendicular line?
The general linear equation in the slope-intercept form is:
y = a*x + b
Where a is the slope and b is the y-intercept.
Let's say that we have another line:
y = c*x + d
These two lines are perpendicular if and only if the slope of one of the lines is equal to the inverse of the opposite of the other line.
So, the two lines:
y = a*x + b
y = c*x + d
Are perpendicular only if c = -(1/a).
Now, we want to find a line perpendicular to:
y=-4x + 16
Then the slope of that perpendicular line must be:
-(1/-4) = (1/4)
So the slope of a line perpendicular to y=-4x + 16 is (1/4).
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The gasoline gauge on a truck reads 1/8 full. after you add 30 gallons the gauge reads 3/4 full. what is the capacity of the tank?
Answers
Answer:
48 gallons
Step-by-step explanation:
\( \frac{3}{4} = \frac{6}{8} \\ \frac{6}{8} - \frac{1}{8} = \frac{5}{8} \\ 5 \div 30 = 6 \\ 6 \times 8 = 48\)
Pick-your-own apples cost $15
per peck. About 30 medium
apples make up a peck. What's
the price per apple?
Answers
Answer:
450
Step-by-step explanation:
because 15 times 30
determine whether the distribution is a discrete probability distribution.
\(
\(\(\(\begin{array}{ll}
X P(x) \\
0 & 0.13 \\
1 & 0.27 \\
2 & 0.18 \\
3 & 0.27 \\
4 & 0.13
\end{array}\)\)\)
\)
Is the probabily distribution a discrete distibution? why?
A-Yes, because the distribution is symmetric.
B- Yes, because the probabilities sum to 1 and are all between 0 and 1 , inclusive.
C- No, beacause some of the probabilities have values greater than 1 or less than 0 .
D- No, because the total probability is not equal to 1
Answers
The true statement about the distribution is (d)- No, because the total probability is not equal to 1
How to determine the true statement?The table of values represents the given parameter
For a probability distribution to represent a discrete probability, the following must be true
The sum of probabilities = 1
Mathematically, this is represented as
∑ p(x) = 1
So, we have
0.13 + 0.27 + 0.18 + 0.27 + 0.13 = 1
Evaluate
0.98 = 1
This is false
Hence, the distribution is a not discrete probability distribution.
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