[tex] \Large \bold{SOLUTION\ 1:} [/tex]
[tex] \small \begin{array}{l} \text{First, we need to check if the given differential} \\ \text{equation is exact.} \\ \\ (1-xy)^{-2} dx + \big[y^2 + x^2 (1-xy)^{-2}\big]dy = 0 \\ \\ \dfrac{dx}{(1-xy)^2} + \left[y^2 + \dfrac{x^2}{(1-xy)^2}\right]dy = 0 \\ \\ \quad M(x, y) dx + N(x, y) dy = 0 \end{array} [/tex]
[tex] \small \begin{array}{l l}\tt\: M(x,y) = \dfrac{1}{(1 - xy)^2}, & N(x,y) = y^2 + \dfrac{x^2}{(1-xy)^2}\\ \\\tt \dfrac{\partial M}{\partial y} = \dfrac{-2(-x)}{(1 - xy)^3}, & \dfrac{\partial N}{\partial x} = \dfrac{2x}{(1 - xy)^2} + \dfrac{-2(-y)x^2}{(1 - xy)^3} \\ \\\tt \dfrac{\partial M}{\partial y} = \dfrac{2x}{(1 - xy)^3}, & \dfrac{\partial N}{\partial x} = \dfrac{2x(1 - xy)+2x^2y}{(1 - xy)^3} \\ \\\tt \: & \dfrac{\partial N}{\partial x} = \dfrac{2x}{(1 - xy)^3} \end{array} [/tex]
[tex] \small \begin{array}{l} \tt\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x} \implies \text{Differential equation is exact.} \\ \\\tt \dfrac{\partial F}{\partial x} = M(x, y) = \dfrac{1}{(1 - xy)^2} \\ \tt\displaystyle F(x, y) = \int \dfrac{1}{(1 - xy)^2} \partial x = -\dfrac{1}{y} \int \dfrac{1}{(1 - xy)^2}(-y)\partial x \\ \\ \tt\:F(x, y) = \dfrac{1}{y(1 - xy)} + h(y) \\ \\ \tt\dfrac{\partial F}{\partial y} = N(x, y) = y^2 + \dfrac{x^2}{(1-xy)^2} \\ \\\tt \dfrac{\partial}{\partial y}\left[\dfrac{1}{y(1 - xy)} + h(y)\right] = y^2 + \dfrac{x^2}{(1-xy)^2} \\ \\ \tt-\dfrac{1 - xy + y(-x)}{y^2(1 - xy)^2} + h'(y) = y^2 + \dfrac{x^2}{(1-xy)^2} \\ \\ \tt-\dfrac{1 - 2xy}{y^2(1 - xy)^2} + h'(y) = y^2 + \dfrac{x^2}{(1-xy)^2} \\ \\ h'(y) = y^2 + \dfrac{x^2}{(1-xy)^2} + \dfrac{1 - 2xy}{y^2(1 - xy)^2} \\ \\ \tt\:h'(y) = y^2 + \dfrac{x^2y^2 - 2xy + 1}{y^2(1-xy)^2} = y^2 + \dfrac{1}{y^2} \\ \\ h(y) = \dfrac{y^3}{3} - \dfrac{1}{y} + C \\ \\ \tt\text{Substituting to }F(x,y),\text{we get} \\ \\ \dfrac{1}{y(1 - xy)} + \dfrac{y^3}{3} - \dfrac{1}{y} = C \\ \\ \quad \quad \text{or} \\ \\ \tt\red{\boxed{\dfrac{x}{1 - xy} + \dfrac{y^3}{3} = C} \Longleftarrow \textit{Answer}} \end{array} [/tex]
[tex] \Large \bold{SOLUTION\ 2:} [/tex]
[tex] \small \begin{array}{l} \tt\text{Since we already know that the equation is exact,} \\ \text{we can then continue solving for the solution by} \\ \text{inspection method or by algebraic manipulation.} \\ \\ \tt(1-xy)^{-2} dx + \big[y^2 + x^2 (1-xy)^{-2}\big]dy = 0 \\ \\ \tt\dfrac{dx}{(1-xy)^2} + \left[y^2 + \dfrac{x^2}{(1-xy)^2}\right]dy = 0 \\ \\ \tt\dfrac{dx}{(1-xy)^2} + y^2 dy + \dfrac{x^2}{(1-xy)^2} dy = 0 \\ \\ \tt\dfrac{dx + x^2dy}{(1-xy)^2} + y^2 dy = 0 \\ \\ \tt\text{Divide both numerator and denominator of the} \\ \tt\text{fraction by }x^2. \end{array} [/tex]
[tex] \small \begin{array}{c}\tt \dfrac{\dfrac{1}{x^2}dx + dy}{\dfrac{(1-xy)^2}{x^2}} + y^2 dy = 0 \\ \tt\\ \tt\dfrac{\dfrac{1}{x^2}dx + dy}{\left(\dfrac{1}{x}-y\right)^2} + y^2 dy = 0 \\ \\ \tt-\dfrac{\left(-\dfrac{1}{x^2}dx - dy\right)}{\left(\dfrac{1}{x}-y\right)^2} + y^2 dy = 0 \\ \\ \tt\displaystyle {\large{\int}} -\frac{d\left(\dfrac{1}{x}-y\right)}{\left(\dfrac{1}{x}-y\right)^2} + \int y^2 dy = \int 0 \\ \\ \tt\implies\tt \dfrac{1}{\dfrac{1}{x}-y} + \dfrac{y^3}{3} = C \\ \\\text{or} \\ \\ \tt\red{\boxed{\dfrac{x}{1 - xy} + \dfrac{y^3}{3} = C} \Longleftarrow \textit{Answer}} \end{array} [/tex]
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Which list of angle measures could be the angle measures of a triange?
Answer:
angles of triangle adds up to 180 degrees
find the circumference with radius (Y) cm?
Answer:
circumference = 2*PI*Y
Step-by-step explanation:
What is an equation of the line that passes through (0, 4) and (2,8)?
Answer:
y=2x+4
Step-by-step explanation:
Hi there!
We are given the points (0, 4) and (2,8)
We want to write the equation of the line that passes through these two points
There are 3 ways to write the equation of the line, but the most common way is with slope-intercept form, which is y=mx+b, where m is the slope and b is the y intercept
So, to use this way, we need to find the slope of the line
The slope can be calculated using the formula [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are points
We are already given the points needed to find the slope, but let's label them to avoid any confusion
[tex]x_1=0\\y_1=4\\x_2=2\\y_2=8[/tex]
Now substitute these values into the formula.
m=[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
m=[tex]\frac{8-4}{2-0}[/tex]
Subtract
m=[tex]\frac{4}{2}[/tex]
Divide
m=2
The slope of the line is 2
Here is the equation of the line so far:
y=2x+b
Now we need to find b
As stated before, b is the y intercept of the line, which is the point in where the line passes through the y axis; the value of x at this point is 0
One of the points we were given earlier is actually the y intercept of the line; that point is (0, 4) *notice how x=0 at this point
The value of b is the value of y at the y intercept; in this case, the value of b is 4
Substitute 4 as b into the equation
y=2x+4
Hope this helps!
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E D 1389 xo С 76.2° B А Find the value of x. O 38.1 O 30.9 69 O 61.8
Based on the intersecting secants theorem, the value of x is: B. 30.9°
What is the Intersecting Secants Theorem?The intersecting secants theorem states that when two secants intersect, the measure of the angle they form is half the difference of the intercepted arcs.
Thus, based on the intersecting secants theorem, we would have the following:
x = 1/2(138 - 76.2)
x = 1/2(61.8)
x = 30.9°
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Shoe Size
2
4
6
8
10
12
At soccer practice, Mike asked each of his teammates to write down their shoe size.
What is the interquartile range for Mike's teammates' shoe sizes?
Answer:
2
Step-by-step explanation:
To find the interquartile range, subtract the lower quartile from the upper quartile.
The upper quartile is 7.
The lower quartile is 5.
Subtract.7–5=2
The interquartile range for Mike's teammates' shoe sizes is 2 sizes.
The interquartile range for Mike's teammates' shoe sizes is R = 2
What in Interquartile Range ( IQR )?The distance between the upper and lower quartiles is known as the interquartile range. Half of the interquartile range corresponds to the semi-interquartile range. Finding the values of the quartiles in a small data set is straightforward.
The spread of the data, or statistical dispersion, is measured by the interquartile range. The middle 50%, fourth spread, or H-spread are further names for the IQR. It is described as the spread between the data's 75th and 25th percentiles.
Given data ,
Let the interquartile range be represented as R
Now , the equation will be
Let the shoe sizes be represented as set A
A = { 2 , 4 , 6 , 8 , 10 }
Now , the upper range of the quartile is = 7
The lower range of the quartile is = 5
And , the interquartile range R = upper quartile - lower quartile
On simplifying , we get
The interquartile range R = 7 - 5 = 2
Hence , the interquartile range R = 2
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PLEASE HELP I WILL GIVE BRAINLIEST
Answer:
20 inches²
Step-by-step explanation:
perimeter = 8 + 3 + 5 + 4
= 20 inches²
solve this problem pls
Answer:
1) 200.
Step-by-step explanation:
Men = 1/³
Women = ⅔
50 men, 100 women
2:1 women to men.
there are 200 people in total.
Answer:
total people : 600
Explanation:
Let the total people be x,
men : [tex]\sf \dfrac{1}{3} (x)[/tex]
women : [tex]\sf \dfrac{2}{3} (x)[/tex]
[tex]\hookrightarrow \sf \dfrac{2}{3} (x) - \sf \dfrac{1}{3} (x) = 200[/tex]
[tex]\hookrightarrow \sf \dfrac{1}{3} (x) = 200[/tex]
[tex]\hookrightarrow \sf x = 600[/tex]
There are total 600 people.
men : (1/3)*600 = 200women : (2/3) * 600 = 400Young Sally went Trick or Treating last Halloween, In her treat bag were 15 pieces of gum, 7 lollipops, 8 tootsie rolls, 6 candy bars, and an apple.
a) Candy bars are her favorite. What percent of her treats are candy bars?
b) She really does not like gum or apples, What percent of the treats things she does not like?
Answer:
A: 16.2% (if you need it rounded, it is 16%)
B: 43.2% (if you need it rounded it is 43%)
Step-by-step explanation:
We need to find the candy bar percentage rate so we divide the amount of candy bars, by the total about of treats collected.
15 + 7 + 8 + 6 + 1 = 37
Now we divide the cady bars by the total amount:
6/37 = 16.2%
Now we do the same thing again but this time with the apples and gum:
16/37 = 43.2%
You did it!
Have a great day!!
Please rate and mark brainliest!!
Find the larger solution of the exponential
equation
2^x^2= 4^3x
Answer:The larger solution is 6
Step-by-step explanation:
Brainliest if correct
Answer:
6.6072 >6.6>6.09>6.009
Step-by-step explanation:
I believe that is it
1) Find the lateral surface area of the rectangular prism in centimeters
A) 96 cm?
B) 108 cm
126 cm
D) 142 cm?
Answer:
It is B because
Step-by-step explanation:
3 x 9 x 4 = 108
The number is greater than 40. The numberis NOT on a red mailbox. The number has more ones than tens.
Answer:
89
Step-by-step explanation:
Based on the clues given, Meg's mailbox number is equal to 89.
The clues adopted by Meg.
In this scenario, the three clues that were adopted by Meg include:
The number is greater than 40.
The number is NOT on a red mailbox.
The number has more ones than tens.
From the image, the numbers that are greater than 40 include:
Green Mail box = 60.
Red Mail Box = 58.
Blue Mail Box = 89.
Red Mail Box = 73.
From the image, the number that is not on a red mailbox are:
Green Mail box = 60.
Blue Mail Box = 89.
From the image, the number that has more ones than tens is:
Blue Mail Box = 89.
By using place value, 8 represents tens while 9 represents ones (unit). Therefore, Meg's mailbox number is equal to 89.
What is the area of this figure?
[tex]a \: = \: square \: area \: + \: triangle \: area \: + \: rectangle \: area \: \\ [/tex]
[tex]a \: = (3 \times 3) + ( \frac{1}{2} \times 3 \times 11) + (3 \times 11) \\ [/tex]
[tex]a = 9 + \frac{33}{2} + 33 \\ [/tex]
[tex]a = 9 + 16.5 + 33[/tex]
[tex]a = 25.5 + 33[/tex]
[tex]a = 58.5 \: \: {ft}^{2} [/tex]
Malcolm draws this map of his neighborhood . Help please and thank you!!!!!
Answer:
Alden, perpendicular, Elm, parallel
Step-by-step explanation:
Write the first three terms of the sequence whose general term is an=(3n)!.
Answer:
3, 6, 9
Step-by-step explanation:
an=3n
a(1)=3(1)= 3
a(2)=3(2)= 6
a(3)=3(3)= 9
Click an item in the list or group of pictures at the botto
correct position in the answer box. Release your mouse
the item to the trashcan. Click the trashcan to clear all y
Rationalize the denominator and simplify.
Answer:
multiply the top number then divide the bottom
Step-by-step explanation:
5 divided by 62.55 need help asap due today and also i cant watch ads
three less than the product of two and a number y
can someone plssss translate that into a algebraic expression
plsss ill give you brainiac
Answer:
[tex]2y-3[/tex]
Step-by-step explanation:
It is saying to subtract 3 from the product of 2 and the variable, y.
multiply the two plynomials
(x + 3)( + 9)
(x + 2)(x + 9)
(x - 8)(x - 6)
im really looking for ways to solve it rather than the solution because it would make my hw alot easier. thanks!
Answer:
1. x² + 12x + 27
2. x² + 11x + 18
3. x² + -14x + 48
Step-by-step explanation:
When you multiply polynomials, you use FOIL.
First
Outer
Inner
Last
For example, to solve the first polynomial (x + 3)(x + 9), you would multiply the first term in each parenthesis to each other. In this case, it would be x · x, which would be equal to x². The next step would be to multiply the first term in the first parenthesis to the last term in the second parenthesis, which are the outer terms. You get 9x when you multiply x by 9. Then, you multiply the inner terms of the polynomial, which is the last term of the first parenthesis and the first term of the second parenthesis. You get 3x. The last part of FOIL is to multiply the last term in each parenthesis to each other. You would multiply 9 and 3 and get 27. Your answer would be x² + 9x + 3x + 27. Simplify the equation to get x² + 12x + 27. That is your final answer. Do the same thing to the other equations to find the answer. (MAKE SURE TO CHECK THE SIGN OF THE TERMS YOU ARE MULTIPLYING FIRST. )
I hope this helped and please mark me as brainliest!
Consider functions f and g
f(x)=(1=2)^×+1 g(x)= log(x - 1) +1 Using three iterations of successive approximation, what is the approximate solution where f(x) g(x)? Use the graph as a starting point.
Answer:
x=39/16
Step-by-step explanation:
I looked it up. Here's the source's explanation;
[tex]f(x)=(1/2)^x+1[/tex]
[tex]g(x)=log(x-1)+1[/tex]
[tex]f(x)=g(x)[/tex]
[tex](1/2)^x +1=log(x-1)+1[/tex]
[tex](1/2)^x=log(x-1)[/tex]
2^(-x)=log (x-1)
x=2.5018
x=39/16
Answer:
41/16
Step-by-step explanation:
took the text and got 100, good luck besties
In the grocery store, eggs are sold in cartons of 12. Write five
ordered pairs made from the corresponding terms of these
two patterns, based on selling 0 to 4 cartons of eggs.
Pattern A: number of cartons sold
Pattern B: number of eggs sold
Plot the points on the plane to the right.
Describe the relationship between the coordinates of the ordered pairs.
Explain how you could find the distance between points (3,6) and (7,6) on the coordinate plane
Answer:
Count the point on the coordinate plane to the point you want to find the distended from Still stuck? Get 1-on-1 help from an expert tutor now.
Step-by-step explanation:
Suppose account numbers for an internet service provider consist of alpha-numeric characters. The first character is a digit (0 through 9). The next 2 characters are capital letters (A through Z) and the last four characters are digits. How many account numbers are possible if no digit appears more than once? Your Answer:
Answer:
20,442,240 account numbers are possible.
Step-by-step explanation:
The first character has 10 possible values.
The next 2 has 26 * 26 = 676 possible combinations.
The last 4 has 9*8*7*6 = 3024 possible combinations.
So the answer is 10 * 676 * 3024
= 20,442,240.
There exist 20,442,240 possible combinations if no digit appears more than once.
How to estimate the account numbers that are possible if no digit appears more than once?The first character contains 10 possible values.
The next 2 contains 26 [tex]*[/tex] 26 = 676 possible combinations.
The last 4 have 9 [tex]*[/tex] 8 [tex]*[/tex] 7 [tex]*[/tex] 6 = 3024 possible combinations.
The has 10 [tex]*[/tex] 676 [tex]*[/tex] 3024 = 20,442,240 possible combinations.
Therefore, there are 20,442,240 possible combinations if no digit appears more than once.
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perform the following division: 3 3/4 ÷ 2/8
Answer:
15
Step-by-step explanation:
[tex]3\frac{3}{4} / \frac{2}{8} = \frac{15}{4} / \frac{2}{8} = \frac{15}{4} * \frac{8}{2} = \frac{120}{8} = 15[/tex]
How come the π/6 is equal to 1/2 and radical 3 over 2?
Answer:
cos(π/6)= radical 3 over 2 sin(π/6) = 1/2
Step-by-step explanation:
In the unit circle the coordinates of the point for the particular angle can be expressed by cos(x) for the x value and sin(x) for the y value. For π/6, the point's x value is radical 3 over 2 as expressed by cos(π/6) and the y value is 1/2 as expressed by sin(π/6)
What is the value of x?
Answer:
lj= lk din 30 give lj= 4✓2
and ml=x= lj cos 45
on solving x= 4✓2 cos45=4
PLEASE help me. Please. Which debates among historians over the effects of the Vietnam War did you discover in your research?
Answer:
The Domino Theory suggested that one nation becoming Communist makes it more likely that others will too. Without American involvement, the South would quickly have fallen to the North and the nation, united under Communism, would have joined the Communist bloc.
Step-by-step explanation:
On the same coordinate plane mark all points (x y) that satisfy the rule y=3x-2 SHOW GRAPH
Answer:
*View the attached graph*
Step-by-step explanation:
Hope it helps!
Need help with this geometric question ASAP, Thank you
Answer:
a. x = 9
b. GH = 30; GK = 20
Step-by-step explanation:
An angle bisector in a triangle divides the sides proportionally.
a.In this triangle, that means ...
(3x +3)/18 = (3x -7)/12
2(3x +3) = 3(3x -7) . . . . . . . multiply by 36
6x +6 = 9x -21 . . . . . . . . . eliminate parentheses
27 = 3x . . . . . . . . . . . . . . add 21 -6x
x = 9 . . . . . . . . . . . . . . . divide by 3
__
b.The segment lengths are found by substituting the value of 3x in each of the given expressions:
GH = 3x +3 = 27 +3 = 30
GK = 3x -7 = 27 -7 = 20
The sum of 9 times a number and 8 is 3
Answer: 9n+8=3
Step-by-step explanation:
Let's break this down
"sum" refers to addition so we must be adding 9 times a number and 8
So far we have "9 times a number + 8"
We can use the variable, n to represent "a number"
Now we have "9 times n +8"
"times" refers to multiplication, so we should multiply 9 and n
Now we have "9n+8"
"is 3" simply means that this will equal 3
So now we have the equation 9n+8=3