(X^2+y^2+x)dx+xydy=0
Solve for general solution
Answers
Check if the equation is exact, which happens for ODEs of the form
\(M(x,y)\,\mathrm dx+N(x,y)\,\mathrm dy=0\)
if \(\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\).
We have
\(M(x,y)=x^2+y^2+x\implies\dfrac{\partial M}{\partial y}=2y\)
\(N(x,y)=xy\implies\dfrac{\partial N}{\partial x}=y\)
so the ODE is not quite exact, but we can find an integrating factor \(\mu(x,y)\) so that
\(\mu(x,y)M(x,y)\,\mathrm dx+\mu(x,y)N(x,y)\,\mathrm dy=0\)
is exact, which would require
\(\dfrac{\partial(\mu M)}{\partial y}=\dfrac{\partial(\mu N)}{\partial x}\implies \dfrac{\partial\mu}{\partial y}M+\mu\dfrac{\partial M}{\partial y}=\dfrac{\partial\mu}{\partial x}N+\mu\dfrac{\partial N}{\partial x}\)
\(\implies\mu\left(\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}\right)=M\dfrac{\partial\mu}{\partial y}-N\dfrac{\partial\mu}{\partial x}\)
Notice that
\(\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}=y-2y=-y\)
is independent of x, and dividing this by \(N(x,y)=xy\) gives an expression independent of y. If we assume \(\mu=\mu(x)\) is a function of x alone, then \(\frac{\partial\mu}{\partial y}=0\), and the partial differential equation above gives
\(-\mu y=-xy\dfrac{\mathrm d\mu}{\mathrm dx}\)
which is separable and we can solve for \(\mu\) easily.
\(-\mu=-x\dfrac{\mathrm d\mu}{\mathrm dx}\)
\(\dfrac{\mathrm d\mu}\mu=\dfrac{\mathrm dx}x\)
\(\ln|\mu|=\ln|x|\)
\(\implies \mu=x\)
So, multiply the original ODE by x on both sides:
\((x^3+xy^2+x^2)\,\mathrm dx+x^2y\,\mathrm dy=0\)
Now
\(\dfrac{\partial(x^3+xy^2+x^2)}{\partial y}=2xy\)
\(\dfrac{\partial(x^2y)}{\partial x}=2xy\)
so the modified ODE is exact.
Now we look for a solution of the form \(F(x,y)=C\), with differential
\(\mathrm dF=\dfrac{\partial F}{\partial x}\,\mathrm dx+\dfrac{\partial F}{\partial y}\,\mathrm dy=0\)
The solution F satisfies
\(\dfrac{\partial F}{\partial x}=x^3+xy^2+x^2\)
\(\dfrac{\partial F}{\partial y}=x^2y\)
Integrating both sides of the first equation with respect to x gives
\(F(x,y)=\dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3+f(y)\)
Differentiating both sides with respect to y gives
\(\dfrac{\partial F}{\partial y}=x^2y+\dfrac{\mathrm df}{\mathrm dy}=x^2y\)
\(\implies\dfrac{\mathrm df}{\mathrm dy}=0\implies f(y)=C\)
So the solution to the ODE is
\(F(x,y)=C\iff \dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3+C=C\)
\(\implies\boxed{\dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3=C}\)
Related Questions
Melanie’s smartphone has 48 apps. This is 4 times the number of apps Priscilla and Cynthia have altogether on their phones. If Cynthia has 5 apps, how many apps does Priscilla have?
Complete the steps below to solve for the number of apps on Priscilla’s smartphone.
A -Write a numerical expression to represent the number of apps Priscilla and Cynthia have combined. (Do not simplify the expression just yet.)
B -Cynthia has 5 apps. Write a numerical expression that represents the number of apps Priscilla has. Refer to the expression you wrote in part A.
C -Simplify the expression from part B to find the number of apps Priscilla has.
D -You just worked with numerical expressions. Now let's try an equation. Let p be Priscilla’s number of apps. Write an equation in terms of p to represent the problem.
E -Solve the equation you found in part D. What is the value of p? Show your work.
F -You solved the problem using two methods: numerical expressions and an equation with variable p. Did you use the same sequence of operations in both methods?
H -What does your answer to part F imply? Explain.
Answers
B. 48/4 -5
C. 7
D. 48=4(5+p)
E. 12=5+p
7=p
F. Yes
H.
PLEASE HELP
Trials in an experiment with a polygraph include 98 results that include 23 cases of wrong results and 75 cases of correct results. Use a 0.05 significance level to test the claim that such polygraph results are correct less than % of the time. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method. Use the normal distribution as an approximation of the binomial distribution.
Answers
what's the formula and slope ?
Answers
Answer:
Formula :\(M= \frac{y2-y1}{x2-x1}\)
\(M= \frac{110- 22}{14- 3}\\\)
Slope/Answer : 8
Find the integer closest to √42.3
Answers
Answer:
42 or 21.15?
not 100% sure but hope this helps
---------------------------------------------
Answer:
7
Step-by-step explanation:
the proper answer is 6.50384501660364
however, this isn't an integer.
to get it to become an integer, we round it up;
6.5 rounded up is 7
and therefore, the answer would be 7
hope this helps
The sum of the first 20 terms of an arithmetic is 50, and the sum of the next 20 terms is -50. Find the is first term and command difference of the sequence?
Answers
Answer:
\(a_1=\frac{39}{8}; \ d=-\frac{1}{4}.\)
Step-by-step explanation:
1) if the first term is 'a₁', the difference of the sequence is 'd', then it is possible to write two equations for the sum of the first 20 terms and the next 20 terms;
2) for the first 20 terms: (a₁+a₂₀)*20/2=50;⇔ (a₁+a₁+19d)*10=50; ⇔2a₁+19d=5;
for the next 20 terms: (a₂₁+a₄₀)*20/2= -50;⇔ (a₁+20d+a₁+39d)*10=-50;⇔ 2a₁+59d= -5.
3) if to solve the system of two equations, then:
\(\left \{ {{2a_1+19d=5} \atop {2a_1+59d=-5}} \right. \ => \ \left \{ {{a_1=\frac{39}{8} } \atop {d=-\frac{1}{4} }} \right.\)
4) finally: the first term is '39/8', the difference is '-1/4'.
Which transformation represents a reflection over the y = × line?
A. (x, y) - (-x, y)
B. (x, y) -+ (-x, -y)
C. (x,y) → (y, x)
D. (x, y) -+ (y, -x)
Answers
The transformation represents a reflection over the y = × line.
A. (x, y) → (-x, y)
A reflection over the y-axis is a transformation that flips a point or shape across the vertical line y = 0.
This means that points on the right side of the y-axis will be reflected to the left side, and vice versa.
Let's examine each option to determine which one represents a reflection over the y-axis.
A. (x, y) → (-x, y):
This transformation reflects the point across the y-axis.
For example, if we have a point (3, 2), after applying this transformation, it becomes (-3, 2).
This represents a reflection over the y-axis.
B. (x, y) → (-x, -y):
This transformation not only reflects the point across the y-axis but also flips it vertically.
For example, if we have a point (3, 2), after applying this transformation, it becomes (-3, -2).
This represents a reflection over the y-axis.
C. (x, y) → (y, x):
This transformation swaps the x and y coordinates of a point, which does not represent a reflection over the y-axis.
Instead, it represents a 90-degree rotation of the point.
D. (x, y) → (y, -x):
This transformation swaps the x and y coordinates of a point and negates the new x-coordinate.
It does not represent a reflection over the y-axis.
Instead, it represents a 90-degree rotation of the point in the counterclockwise direction.
Based on the explanations above, both options A and B represent a reflection over the y-axis.
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A teapot,which has a capacity of one and half litters, can fill six identical mugs with tea.what is the capacity of the six mugs altogether?
Answers
Answer:250
Step-by-step explanation:
Will pick brainliest! I need help with this, actual effort in answering is much appreciated.
Answers
Answer:
option 2
Step-by-step explanation:
4^2=16/8=2. 4^2=16/16=1. 2-1=1
What is √-11 in the form a+bi?
Answers
Given:
The number is \(\sqrt{-11}\).
To find:
The a+bi form of given number.
Solution:
We have,
\(\sqrt{-11}\)
It can be written as
\(\sqrt{-11}=\sqrt{-1\times 11}\)
\(\sqrt{-11}=\sqrt{-1}\times \sqrt{11}\) \([\because \sqrt{ab}=\sqrt{a}\sqrt{b}]\)
\(\sqrt{-11}=i\times \sqrt{11}\) \([\because \sqrt{-1}=i]\)
\(\sqrt{-11}=\sqrt{11}i\)
Here, real part is missing. So, it can be taken as 0.
\(\sqrt{-11}=0+\sqrt{11}i\)
So, a = 0 and \(b=\sqrt{11}\).
Therefore, the a+bi form of given number is \(0+\sqrt{11}i\).
There are how many halves in 2 1/2
Answers
Answer:
there are 5 halves in 2 1/2
Answer:
5
Step-by-step explanation:
1/2 +1/2 = 1
Please help quick!!!!
Answers
Answer is Option B
If any doubt leave a comment
NEED HELP ASAP!! What is the probability that either event will occur?
Answers
The value of probability P (A or B) is,
⇒ P (A or B) = 48
We have to given that,
The Venn diagram of the probability is shown in image.
Now, By given diagram,
⇒ P (A) = 24 + 14
⇒ P (A) = 38
⇒ P (B) = 24 + 10
⇒ P (B) = 34
⇒ P (A and B) = 24
We have to given that,
The expression is,
P (A or B) = P (A) + P (B) - P (A and B)
Substitute all the values, we get;
P (A or B) = 38 + 34 - 24
P (A or B) = 48
Thus, The value of probability P (A or B) is,
⇒ P (A or B) = 48
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If triangles ABC and DEF are similar, what is y? Show your work.
Answers
The value of y is 18
What are similar triangles?Similar triangles have the same corresponding angle measures and proportional side lengths. The angles of the two triangle must be equal and it not necessary they have equal sides.
Therefore the corresponding angles of similar triangles are congruent and the ratio of corresponding sides of similar triangles are equal.
Therefore;
14/21 = 12/y
14y = 21 × 12
14y = 252
divide both sides by 14
y = 252/14
y = 18
Therefore the value of y is 18.
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Someone please help ASAP will mark as Brainliest
Answers
About 217,000 high school students took the AP Statistics exam in 2017. The free-response section of the exam consisted of five open-ended problems and an investigative task. Each free-response question is scored on a 0 to 4 scale (with 4 being the best). For one of the problems, a random sample of 30 student papers yielded a mean score of x =1.267 and a standard deviation of 1.230. a. Find and interpret the standard error of the mean. b. Construct and interpret a 90% confidence interval to estimate the true mean score on this question.
Answers
The standard error is of 0.2246, which means that the mean scores for samples of 30 vary around 0.2246 from the mean.
What is a confidence interval?The confidence interval is the range of values that you expect your estimate to fall between a certain percentage of the time if you run your experiment again or re-sample the population in the same way.
The confidence interval formula is \(\bar x\pm t\frac{s}{\sqrt{n}}\).
Where, \(\bar x\) is the sample mean, t is the critical value, n is the sample size and s is the standard deviation for the sample.
Here,
\(\bar x\) =1.267, s=1.23, n=30
The standard error is Se= 1.23/√30
= 0.2246
Item a:
The standard error is of 0.2246, which means that the mean scores for samples of 30 vary around 0.2246 from the mean.
Item b:
Using a t-distribution calculator, considering a confidence level of 0.99 with 30 - 1 = 29 df, the critical value is t = 2.7564.
\(\bar x\pm tS_c\)
\(\bar x+ tS_c\) =1.267-2.7564(0.2246) =0.6479
\(\bar x- tS_c\) =1.267+2.7564(0.2246) =1.8861
Therefore, the 99% confidence interval to estimate the true mean score on this question is (0.6479, 1.8861). It means that we are 99% that the true mean score of all students in this question is between 0.6479 and 1.8861.
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a student spends 18 out of 35 of his pocket money on transport and fruit what is the fraction left?
Answers
To find the fraction of pocket money left after spending on transport and fruit, we need to subtract the amount spent from the total pocket money and express it as a fraction.
The student spends 18 out of 35 of his pocket money, which means he has (35 - 18) = 17 units of his pocket money left.
Therefore, the fraction of pocket money left can be written as 17/35.
Find the value of X.
Answers
Answer:
148 degrees
Step-by-step explanation:
2 Remote interior angle added to one another equals the measure of the exterior angle
44+104=148
3
\(3 \sqrt{81} \)
what's the answer
Answers
Answer will be 27.
Given,
3√81
Now, to solve the expression the squares of whole numbers and square roots for some numbers must be known.
For example, squares of
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
Square roots,
√100 = 10
√81 = 9
√64 = 8
√49 = 7
√36 = 6
√25 = 5
√16 = 4
√9 = 3
√4 = 2
√1 = 1
Now ,
3√81 = 3× 9
= 27.
Thus the value is 27.
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The table shows information about the masses of some dogs.
a) Work out the minimum number of dogs that could have a mass of more than 24 kg.
b) Work out the maximum number of dogs that could have a mass of more than 24 kg.
Answers
a) A minimum of number of dogs that could have a mass of more than 24 kg is 6.
b) The maximum number of dogs that could have a mass of more than 18 kg is 20.
What do minimum and maximum value mean?Rearrange the function using fundamental algebraic concepts to get the value of x when the derivative equals 0. This response gives the x-coordinate of the function's vertex, which is where the maximum or minimum will occur. To find the minimum or maximum, rewrite the solution into the original function.
The minimum of the values present in the interval 20 to 40 is 6
The maximum of the values present in the interval 20 to 40 is (12+6) = 18
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Question 2 (1 point) (06.04 MC) Find the product of (x − 5)2. Question 2 options: x2 + 10x + 25 x2 − 10x + 25 x2 − 25 x2 + 25
Answers
\( { (x - 5) }^{2} = {x}^{2} - 10x + 25\)
29. Assertion :7√5, √2+21 are the irrational number. Reason: every integer is an rational number 30.
Answers
The assertion is true. Both 7√5 and √2 + 21 are irrational numbers because they cannot be expressed as fractions.
The assertion is true. Both 7√5 and √2 + 21 are irrational numbers.
An irrational number is defined as a number that cannot be expressed as a fraction of two integers and has an infinite non-repeating decimal representation.
In the case of 7√5, the square root of 5 is an irrational number because it cannot be expressed as a fraction. Multiplying it by 7 does not change its irrational nature.
Similarly, √2 is also an irrational number because the square root of 2 cannot be expressed as a fraction. Adding 21 to √2 does not alter its irrationality.
The reason provided, that every integer is a rational number, is not relevant to the given assertion. While it is true that every integer is a rational number because it can be expressed as a fraction (e.g., 3 can be written as 3/1), it does not contradict the fact that 7√5 and √2 + 21 are irrational numbers.
In conclusion, the assertion is valid, and both 7√5 and √2 + 21 are irrational numbers.
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A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed. Find the number.
Answers
If 18 is subtracted from the number, the digits are reversed.
Solution:Let the digit in the unit's place be x and the digit at the tens place be y.
Number = 10y + xThe number obtained by reversing the order of the digits is = 10x + y
ATQ:Condition: 110y + x = 6(x + y) + 4
10y + x = 6x + 6y + 4
10y + x - 6x - 6y = 4
- 5x + 4y = 4
5x - 4y = - 4 ............... (1)Condition : 2(10y + x) - 18 = 10x + y
10y + x - 10x - y = 18
- 9x + 9y = 18
- 9(x - y) = 18
x - y = - 18/9
x - y = - 2 ..............(2)On multiplying equation (2) by 4:4x - 4y = -8............(3)
On Subtracting equation (3) from equation (1), we obtain:5x - 4y = - 4
4x - 4y = - 8
(-) (+) (+)
----------------------
x = 4On putting x = 4 in eq (1) we obtain:-5x - 4y = - 4
5(4) - 4y = - 4
20 - 4y = - 4
- 4y = - 4 - 20
- 4y = - 24
y = 24/4
y = 6Now, Number = 10y + x = 10 × 6 + 4 = 60 + 4 = 64
Hence, the number is 6417 The table below shows the distance a car has traveled.
50
20
f
40
Minutes
Distance
Traveled
(in miles)
What is the meaning of the slope of the linear model for the data?
60
100
a) The car travels 5 miles every minute.
b) The car travels 4 miles every minute.
c) The car travels 4 miles every 5 minutes.
d) The car travels 5 miles every 4 minutes.
125
80
100
Answers
Given statement solution is :- None of the given options (a, b, c, or d) match the meaning of the slope. The correct interpretation is that the car travels approximately 0.8 miles every minute.
To determine the meaning of the slope of the linear model for the given data, let's analyze the information provided. The table represents the distance traveled by a car at different time intervals.
Minutes | Distance Traveled (in miles)
50 | 20
20 | f
40 | 60
100 | a
125 | 80
100 | 100
To find the slope of the linear model, we need to calculate the change in distance divided by the change in time. Let's consider the intervals where the time changes by a fixed amount:
Between 50 minutes and 20 minutes: The distance changes from 20 miles to 'f' miles. We don't have the exact value of 'f', so we can't calculate the slope for this interval.
Between 20 minutes and 40 minutes: The distance changes from 'f' miles to 60 miles. Again, without knowing the value of 'f', we can't calculate the slope for this interval.
Between 40 minutes and 100 minutes: The distance changes from 60 miles to 'a' miles. We don't have the exact value of 'a', so we can't calculate the slope for this interval.
Between 100 minutes and 125 minutes: The distance changes from 'a' miles to 80 miles. Since we still don't have the exact value of 'a', we can't calculate the slope for this interval.
Between 125 minutes and 100 minutes: The distance changes from 80 miles to 100 miles. The time interval is 25 minutes, and the distance change is 100 - 80 = 20 miles.
Therefore, based on the given data, we can conclude that the car travels 20 miles in 25 minutes. To determine the meaning of the slope, we divide the distance change by the time change:
Slope = Distance Change / Time Change
= 20 miles / 25 minutes
= 0.8 miles per minute
So, none of the given options (a, b, c, or d) match the meaning of the slope. The correct interpretation is that the car travels approximately 0.8 miles every minute.
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lim (cosec 2x - xcosec³x)
x approaches 0
Answers
The limit lim(x approaches 0) (cosec(2x) - x*cosec³(x)) is undefined.
We have,
To evaluate the limit of the expression as x approaches 0, we can simplify the expression first.
The expression is given as lim(x approaches 0) (cosec(2x) - x cosec³(x)).
Using trigonometric identities, we can rewrite cosec(2x) as 1/sin(2x) and cosec³(x) as 1/(sin(x))³.
Substituting these into the expression,
We get lim(x approaches 0) (1/sin(2x) - x (1/(sin(x))³)).
Now, let's evaluate the limit term by term:
lim(x approaches 0) (1/sin(2x)) = 1/sin(0) = 1/0 (which is undefined).
lim(x approaches 0) (x (1/(sin(x))³))
= 0 (1/(sin(0))³)
= 0 x 1/0 (which is also undefined).
Since both terms of the expression are undefined as x approaches 0, we cannot determine the limit of the expression.
Therefore,
The limit lim(x approaches 0) (cosec(2x) - x*cosec³(x)) is undefined.
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Burns that can be treated at home are most likely
to be
O second- and third-degree burns.
O first-degree burns.
O second-degree burns.
O third-degree burns.
DONE ✔
00000
Answers
Most likely, burns that can be treated at home are first-degree burns. The Option B
Which types of burns can be treated at home?First-degree burns are the mildest form of burns and typically involve damage to the outer layer of the skin. These burns are characterized by redness, pain and swelling but do not typically result in blisters or open wounds.
Since its affect superficial layers of the skin, they can be treated at home with first aid measures. This include rinsing the burn with cool water, taking over-the-counter pain relievers and keeping the burn clean and protected with a non-stick bandage.
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i need the answer to this question
Answers
Answer:
Area of annulus is 40.85cm² to 2d.p
Step-by-step explanation:
Area of shaded part=Area of bigger circle -Area of smaler circle
Area of annulus= πR²- πr²= π(R²-r²)
A=3.142(7²-6²)
A=3.142(49-36)
A=3.142×13
A=40.846cm²
A=40.85cm² to 2d.p
Suppose that a recent poll of American households about pet ownership found that for households with pets, 45% owned a dog, 34% owned a cat, and 10% owned a bird. Suppose that three households are selected randomly and with replacement and the ownership is mutually exclusive. What is the probability that all three randomly selected households own a dog
Answers
Answer: 0.091125
Step-by-step explanation:
Given : P(dog) = 0.45
If 3 households are selected randomly and with replacement and the ownership is mutually exclusive.
Then , probability that all three randomly selected households own a dog = \((0.45)^3=0.091125\)
Hence, required probability = 0.091125
The probability that all three randomly selected households own a dog is
0.091125.
Given that,
Suppose that a recent poll of American households about pet ownership found that for households with pets, 45% owned a dog, 34% owned a cat, and 10% owned a bird.Three households are selected randomly and with replacement and the ownership is mutually exclusive.Based on the above information, the calculation is as follows:
\(= 0.45^3\)
= 0.091125
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Find an equation of the line
Through (-3,-7); parallel to 2x + 3y = 5
Answers
Answer:
2x +3y = -27
Step-by-step explanation:
You want an equation for a line parallel to 2x +3y = 5 and through the point (-3, -7).
Parallel lineA parallel line will have the same slope as the given line, but will have different intercepts. In the standard-form equation given, that means the coefficients of the variables will be the same, but the constant will be different.
To make the constant appropriate for the given point, we use the (x, y) values of the given point to find it:
2x +3y = c
2(-3) +3(-7) = c = -6 -21 = -27
The desired equation is ...
2x +3y = -27
A student invested 1300 in account that pays 4% annual compound interest. He will not make any additional deposits or withdraws. How much will he have in the account at the end of five years?
Answers
Year 2: $1352x0.04 = $54.08 + $1352 = $1406.08
Year 3: $1406.08x0.04 = $56.24 + $1406.08 = $1542.62
Year 4: $1542.62 x 0.04 = $61.70 + $1542.62= $1604.32
Year 5: $1604.32 x 0.04 = $64.17
The Student will have $1668.14 at the end of 5 years. I broke it down by year I hope this helps
Supervisor: "I think you have been doing a great job, but you haven't been signing many people up for our new service feature. I want you to set a goal of signing up 25% of your new customers for our new service feature."
Representative: "If I get 96 new customers that means I have to get __________ of them to sign up for the new service feature."
Answers
It have get \(24\) customer of them to sign up for the new service feature.
Given:
Sign up percentage is \(25\) %.
\(96\) customer.
To find the solution by,
Multiply customer and sign up
\(=96*25\)%
\(=96*\frac{25}{100} \\\\=2400/100\\\\=24\)
Thus, it have get \(24\) customer of them to sign up for the new service feature.
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