Verify that ϕ(x)=2(1−cex), where cis an arbitrary constant, is a one-parameter family of solutions to dydx=y(y−2)2.Graph the solution curves corresponding to c=0,±1,±2using the same coordinate axes.
Answers
Answer:
Following are the solution:
Step-by-step explanation:
Given equation:
[tex]\frac{dx}{dy}= \frac{y(y-2)}{2}........(a)[/tex]
[tex]\Phi (x) = \frac{2}{1-ce^x}[/tex]
In the question [tex]\Phi (x)[/tex], is a solution of the equation (a) only if when [tex]\Phi (x)[/tex]satisfies equation (a):
let us find [tex]\Phi' (x)[/tex]:-
[tex]\Phi(x) =\frac{d}{dx} \Phi (x)=\frac{d}{dx}(\frac{2}{1-ce^x})[/tex]
[tex]= 2 \frac{d}{dx}({1-ce^x}^-1)\\\\= 2 (-1) ({1-ce^x})^{-2} \frac{d}{dx}({1-ce^x})\\\\= -2 ({1-ce^x})^{-2} (-ce^x})\\\\= 2(-ce^x) ({1-ce^x})^{-2} \\\\[/tex]
Now:
[tex]\frac{dy}{dx}=\frac{y(y-2)}{2}\\\\\frac{d}{dx}\Phi (x) =\frac{\Phi (x) (\Phi (x)-2)}{2}\\[/tex]
by subtracting the value of [tex]\Phi (x)[/tex]and [tex]\frac{d}{dx}[/tex][tex]\Phi (x)[/tex],we get:
[tex]\to[/tex][tex]2ce^x(1-ce^x)^-2= \frac{(\frac{2}{1-ce^x})(\frac{2}{1-ce^x})^{-2}}{2}[/tex]
[tex]=2ce^x (1-ce^x)^{-2}=\frac{1}{2}[\frac{2}{1-ce^x}(\frac{2-2(1-ce^x)}{1-ce^x)}]\\\\= \frac{1}{1-ce^x} \times \frac{2ce^x}{1-ce^x}\\\\=\frac{2ce^x}{(1-ce^x)^2}\\\\=2ce^x (1-ce^x)^-2[/tex]
[tex]\bold{2ce^x(1-ce^x)^-2=2ce^x(1-ce^x)^-2}[/tex]
[tex]\Phi (x)[/tex] the solution of the given equation. [tex]\Phi (x)[/tex] has one Parameter Family of
[tex]\frac{dy}{dx}=\frac{y(y-2)}{2}[/tex]
[tex]\Phi (x) =\frac{2}{1-ce^x}\\\\_{when} \ \ \ c= 0 \to \Phi (x) =\frac{2}{1-0\times e^x}=2\\\\_{when} \ \ \ c= 1 \to \Phi (x) =\frac{2}{1-1\times e^x}=\frac{2}{1-e^x}\\\\_{when} \ \ \ c= -1 \to \Phi (x) =\frac{2}{1-(-1)\times e^x}=\frac{2}{1+e^x}\\\\_{when} \ \ \ c= 2 \to \Phi (x) =\frac{2}{1-(-2)\times e^x}=\frac{2}{1-2e^x}\\\\_{when} \ \ \ c= -2 \to \Phi (x) =\frac{2}{1-(-2)\times e^x}=\frac{2}{1+2e^x}\\\\[/tex]
Related Questions
A normal population has the mean of 10 and the variance of 25 . A random sample of size n-28 is selected.
(a) Find the standard deviation of the sample mean. Round your answer to two decimal places (e.g. 98.76)
(b) How large must the sample be if you want to halve the standard deviation of the sample mean?
Answers
Answer:
Step-by-step explanation:
Given that:
population mean = 10
variance [tex]\sigma^2[/tex] = 25 ; [tex]\sigma[/tex] =[tex]\sqrt{ 25}[/tex] = 5
sample size n = 28
The standard deviation of the sample mean:
= [tex]{\dfrac{5}{\sqrt{28}}[/tex]
= 0.95
To halve the standard deviation of the sample mean, the size of how large the sample will be can be computed as follows:
[tex]\dfrac{sd_1}{sd_2} = \dfrac{\dfrac{\sigma}{\sqrt{n}} }{\dfrac{\sigma}{\sqrt{n}} }[/tex]
[tex]\implies \dfrac{\dfrac{5}{\sqrt{28}} }{\dfrac{5}{\sqrt{28}} }[/tex]
[tex]= \dfrac{5}{\sqrt{28}} \times \dfrac{\sqrt{28}}{5}[/tex]
= 2
n = 4 × 28
n = 112
The standard deviation is 0.95 and if the standard deviation is halved then the sample mean is 112.
What is a standard deviation?It is the measure of the dispersion of statistical data. Dispersion is the extent to which the value is in a variation.
A normal population has a mean of 10 and a variance of 25.
A random sample of sizes n-28 is selected.
A. The standard deviation will be
[tex]\sigma = \dfrac{\sqrt{Var(x)}}{n} \\\\\sigma = \dfrac{\sqrt{25}}{\sqrt{28}} \\\\\sigma = 0.95[/tex]
B. The sample be if you want to halve the standard deviation of the sample mean
[tex]\rm \dfrac{SD_1}{SD_2} = 2 = \dfrac{\sqrt{n}}{\sqrt{28}}\\\\n = 4*28 \\\\n= 112[/tex]
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Find the equation of the tangent line to the curve at the given point?
y = x cot(x) at the point with x-coordinate= π/4
Answers
Answer:
The equation of the tangent line
[tex]y - \frac{\pi }{4} = (\frac{-\pi }{2} +1)( x - \frac{\pi }{4} )[/tex]
Step-by-step explanation:
Step(i):-
Given function y = x cot (x) ....(i)
Differentiating equation (i) with respective to 'x' , we get
[tex]\frac{dy}{dx} = x (-Co sec^{2} (x)) +cot(x) (1)[/tex]
Step(ii):-
The slope of the tangent line
[tex]\frac{d y}{d x} = -x Co-sec^{2} (x) +cot x[/tex]
[tex](\frac{d y}{d x} )x_{=\frac{\pi }{4} } = -\frac{\pi }{4} Co-sec^{2} (\frac{\pi }{4} ) +cot \frac{\pi }{4}[/tex]
We will use trigonometry formulas
[tex]Cosec(\frac{\pi }{4} ) = \sqrt{2}[/tex]
[tex]sec(\frac{\pi }{4} ) = \sqrt{2}[/tex]
[tex]Cot(\frac{\pi }{4} ) = 1[/tex]
Now the slope of the tangent
[tex]\frac{dy}{dx} =-\frac{\pi }{4} (\sqrt{2})^{2} )+1[/tex]
[tex]\frac{dy}{dx} =-\frac{\pi }{2} +1[/tex]
Step(iii):-
Given
Substitute [tex]x=\frac{\pi }{4}[/tex] in y = x cot (x)
[tex]y = \frac{\pi }{4} cot (\frac{\pi }{4} )[/tex]
[tex]y = \frac{\pi }{4}[/tex]
The point of the tangent line [tex](x ,y ) = (\frac{\pi }{4} , \frac{\pi }{4} )[/tex]
The equation of the tangent line
[tex]y - y_{1} = m ( x - x_{1} )[/tex]
[tex]y - \frac{\pi }{4} = (\frac{-\pi }{2} +1)( x - \frac{\pi }{4} )[/tex]
Which set of numbers could be the side lengths of a right triangle?
A. 3,4,6
B.4,5,7
C. 5, 12, 13
D. 7. 20, 21
E. 8, 15, 16
Answers
4(2x+6)=5(x+5)+2 Infinite solutions Infinite solutions No solution No solution One solution; x=0One solution; x is equal to 0 One solution; x=1
Answers
Answer:
x = 1
Step-by-step explanation:
Solve for x:
4 (2 x + 6) = 5 (x + 5) + 2
Hint: | Distribute 5 over x + 5.
5 (x + 5) = 5 x + 25:
4 (2 x + 6) = 5 x + 25 + 2
Hint: | Group like terms in 5 x + 2 + 25.
Grouping like terms, 5 x + 2 + 25 = 5 x + (25 + 2):
4 (2 x + 6) = 5 x + (25 + 2)
Hint: | Evaluate 25 + 2.
25 + 2 = 27:
4 (2 x + 6) = 5 x + 27
Hint: | Write the linear polynomial on the left hand side in standard form.
Expand out terms of the left hand side:
8 x + 24 = 5 x + 27
Hint: | Move terms with x to the left hand side.
Subtract 5 x from both sides:
(8 x - 5 x) + 24 = (5 x - 5 x) + 27
Hint: | Combine like terms in 8 x - 5 x.
8 x - 5 x = 3 x:
3 x + 24 = (5 x - 5 x) + 27
Hint: | Look for the difference of two identical terms.
5 x - 5 x = 0:
3 x + 24 = 27
Hint: | Isolate terms with x to the left hand side.
Subtract 24 from both sides:
3 x + (24 - 24) = 27 - 24
Hint: | Look for the difference of two identical terms.
24 - 24 = 0:
3 x = 27 - 24
Hint: | Evaluate 27 - 24.
27 - 24 = 3:
3 x = 3
Hint: | Divide both sides by a constant to simplify the equation.
Divide both sides of 3 x = 3 by 3:
(3 x)/3 = 3/3
Hint: | Any nonzero number divided by itself is one.
3/3 = 1:
x = 3/3
Hint: | Any nonzero number divided by itself is one.
3/3 = 1:
Answer: x = 1
c
Which equation could you use to solve for x in the proportion 5
- 2
Vx
4X= 14
4x=45
5x= 13
5x = 36
ANSWER QUICK!!
Answers
Answer:
4x= 14
Step-by-step explanation:
This equation will be in standard form if you subtract 6x2 from
both sides, add 30 to both sides and subtract ex from both
sides.
6x2 + 2x - 30 = 4x2 +9x
A.True
B.False
Answers
The answer to it is true
Which of the following sets are equal? A = {2, 3, 4} n B = {XER 1 4 x < 5} n C = { ERI 1 < x < 5} n D = {xEZ| 1 < x < 59} n E = {xez | 1 < x < 59} n
Answers
Answer:
Step-by-step explanation:
The question is not too clear. check the attachment for better view of the question.
From the options The set A = {2, 3, 4} will be compared to other options to determine which of the remaining set is equivalent to the set A.
x e R means the values in the set are real numbers and real numbers can either be rational or irrational. R= Hence the real numbers between 1 and 5 are not restricted to only positive integer values but also decimal numbers. and since set A does nit contain ant rational number, hence both set B and C are not equivalent to set A.
Also, set B is not equivalent to set C even though the values in both sets are integers. They are not equal because 1 is included in set B but not in set C.
Set D and E are equivalent because they both contains integers between 1 and 5. Integers are positive whole numbers between 1 and 5 since the both sets doesn't contain any negative value. Hence the set A = set D = set E
find the absolute value of -0.909?
Answers
0.909 is the absolute value of -0.909
What is Number system?A number system is defined as a system of writing to express numbers.
The given number is a decimal number
-0.909
Minus zero point nine zero nine
We need to find the absolute value of -0.909
Absolute value of a Decimal is its numeric value without its sign
The absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign.
0.909 is the absolute value of -0.909
Hence, 0.909 is the absolute value of -0.909
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I need help with d ASAP plz explain
Answers
Answer:
a. g(0)= 1
b. f(-2)= 0
d. g(x-2)= 5x-9
Step-by-step explanation:
a. g(x)=5x+1 *substitute 0 for x*
g(0)= 5(0)+1
g(0)= 1
b. f(-2)= [tex](-2)^2-4[/tex] *substitute -2 for x*
f(-2)= 4-4 =0
c. g(x-2)= 5(x-2)+1 *substitute x-2 for x*
g(x-2)= 5x-10+1 *distributive property*
g(x-2)=5x-9 *simplify*
Find the slope of the given line, if it is defined. 2x + 1 = 0
Answers
Given :
A line 2x + 1 =0 .
To Find :
The slope of the given line .
Solution :
We know , slope of line is given by the tangent of the angle between the x-axis and the line .
Now, for line 2x + 1 =0 i.e [tex]x=\dfrac{-1}{2}[/tex] .
The line is perpendicular to x-axis and cuts the x-axis at [tex]\dfrac{-1}{2}[/tex] .
Therefore , the angle between the line and x-axis is [tex]90^o[/tex] .
So , slope [tex]m=\tan\ 90^o[/tex] i.e undefined .
Therefore , the slope of given line is not defined .
Hence , this is the required solution .
Solve for m.
7
D= (m - 64)
2
==
Answers
=(D-64)2
We move all terms to the left:
-((D-64)2)=0
We calculate terms in parentheses: -((D-64)2), so:
(D-64)2
We multiply parentheses
2D-128
Back to the equation:
-(2D-128)
We get rid of parentheses
-2D+128=0
We move all terms containing D to the left, all other terms to the right
-2D=-128
D=-128/-2
D=+64
sita purchased 100 pencils of rs600. when she sold all the pencils she lost 20%. find the selling price of each pencils.
Answers
Answer:
rs480
Step-by-step explanation:
100%=r600 100%-20%=80%
80%=x
80 x 600 =48000
[tex]\frac{48000}{100}[/tex]=480
answer reeeeeeeeeeeeeeeeeeeeeeeee
Answers
Answer:
36 and 49
Step-by-step explanation:
6x6 and 7x7
Find the product of -3/5 • -1.5.
Answers
Answer:
0.9
Step-by-step explanation:
I am not sure how to really explain but the 2 negatives equal a positive. So if there is 2 Negatives also know as - = a positive. And a positive is just the the opposite of a negative
The product of the given fractions is 3/25.
The given fractions are -3/5 and -1/5.
What is the product of fractions?The product of two fractions can often be expressed by an equivalent fraction where the numerator and denominator have been divided by a common factor. The product of two fractions is the product of the numerators and the product of the denominators.
Now, multiply the given fractions
-3/5 × -1/5
Multiply numerators and write the result in numerator and multiply denominators and write the result in denominator.
That is, -3/5 × -1/5 = 3/25
Therefore, the product of the given fractions is 3/25.
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Lydia owns a food truck that sells tacos and burritos. She only has enough supplies to make 115 tacos or burritos. She sells each taco for $4 and each burrito for $6.75. Lydia must sell at least $680 worth of tacos and burritos each day. If 32 tacos were sold, determine the minimum number of burritos that the Lydia must sell in order to meet the requirements. If there are no possible solutions, submit an empty answer.
Answers
Answer:
Lydia must sell 552 burritos
Step-by-step explanation:
Which of the following would be considered the most conservative alpha level?
a. .01
b. .05
c. .10
d. .15
Answers
Answer:
C.0.1
Step-by-step explanation:
Because this is 10% confidence confirmed
67.9 is ten times as much as what number?
A. 6.79
B. 6,790
C. 607.9
D. 6.7
Answers
Answer:
6.79
Step-by-step explanation:
6.79 x 10 = 67.9
The unknown number z would be 6.79
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.
If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
WE have been given that 67.9 is ten times as much as unknown number;
Let the unknown number be z;
Z x 10 = 67.9
Now divide by 10 on both sides, we get;
Z = 67.9/10
Z = 6.79
Therefore,
6.79 x 10 = 67.9
Hence, the unknown number z would be 6.79
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Identify the line(s) of symmetry for kite ABCD.
les )
A)line BD
B)line AC
C)line EF
D)no line of symmetry
Answers
Answer:Line BD
Step-by-step explanation:
The top is shorter than the bottom, so it can't be AC. EF will only have about half of the section covered. BD is symmetrical so it can't be no lines. So it must be BD.
If a woodchuck could chuck 2
pieces of wood every hour, how big is a beavers dam?
Answers
Answer:
How many hours?
Step-by-step explanation:
the answer to this is 2x
X is the number of hours and you multipy it by 2 wood
3x+2y=4 solve for y?
Answers
The given equation term of y is, y=(3/2)x-2.
What is the equation?A mathematical statement consisting of an equal symbol between two algebraic expressions with the same value is known as an equation.
Given equation in the problem is;
3x+2y=4
To solve the given equation for y first rearrange the equation and put the y variable on the left-hand side as;
2y=3x-4
Divide the whole equation by 2 we get;
y=(3/2)x-2
Hence, the given equation term of y is, y=(3/2)x-2.
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I get paid $40 a month. If i want 5 less each month, how much should i get paid every week?
Answers
Answer:
$5 each week
Step-by-step explanation:
$40 a month
$10 a week
$10-$5 is $5 each week
If you want to be paid $5 less, then every month you will be paid $35
There the amount that you get paid every week will be:
35 ÷ 4 ( 1 month have 4 weeks )
= $8.75
Select all that apply.
Which expressions are equivalent to 6x +18?
6(2+3)
3(2x + 6)
2 (32 +9)
6(x + 18)
3(x+6)
Answers
Hope this helps!
The expression equivalent to 6x + 18 is 3(2x + 6) thus option (B) is correct.
What is an expression?An expression is a combination of some mathematical symbol such that an arithmetic operator and variable such that are all constrained and create an equation.
Expression is very useful to determine the end or root value of constraint.
As per the given,
6x + 18
Take 3 as common,
3(2x + 6)
Hence "The expression equivalent to 6x + 18 is 3(2x + 6)".
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Name 3 rays in this geometric construction
E
A
D
B
Your answer
Answers
Why did I get this question wrong?
Answers
Answer:
9.25
Step-by-step explanation:
M₄ means the area using 4 rectangles with heights equal to the function evaluated at the midpoints.
The width of each rectangle is (4−0)/4 = 1.
The area of each rectangle is:
M₁ = (1) (1.12) = 1.12
M₂ = (1) (1.80) = 1.80
M₃ = (1) (2.69) = 2.69
M₄ = (1) (3.64) = 3.64
The total area is therefore:
M = 1.12 + 1.80 + 2.69 + 3.64
M = 9.25
46 winners divided $33 equally. what decimal is equivalent to the fraction of a dollar that each winner received? round to the nearest hundredths.
Answers
Answer:
1.39
Step-by-step explanation:
Solve for y. 6 x + 2y = z
Answers
Answer:
[tex]y = \frac{z}{2} -3x[/tex]
Step-by-step explanation:
Isolate the variable, y. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.
PEMDAS =
Parenthesis
Exponents & roots
Multiplication
Division
Addition
Subtraction
& is the order of operation.
First, subtract 6x from both sides:
6x (-6x) + 2y = z (-6x)
2y = z - 6x
Next, divide 2 from both sides:
(2y)/2 = (z - 6x)/2
y = (z - 6x)/2
y = (z/2) - 3x
[tex]y = \frac{z}{2} -3x[/tex] is your answer.
~
(30 points) the line contains the point (5,-7) and is parallel to 4x+ 3y =-9 The equation of the line is:
Answers
Answer:
4x +3y = -1
Step-by-step explanation:
To find the new constant for the equation, use the given x- and y-values:
4(5) +3(-7) = 20 -21 = -1
The equation of the parallel line in standard form is ...
4x +3y = -1
Divide rs 21000 between Nisha and Nishan so that Nisha gets double than Nishan
Answers
let the nisha be x and nishan be y
Step-by-step explanation:
ATQ
your statement is x=2y
ILL GIVE YOU BRAINLIST !!
Nick's mom gave him $10 allowance. His little sister was given $4 less than Nick. Write an expression for the amount that Nick's little sister was given
Answers
Answer:
Let x be the amount Nick's little sister is given.
x = 10 - 4
Select all that apply which expressions are equivalent to 6X +12
(A) 6(x+12)
(B) x(6+12)
(C) 3(2x+4)
(D) 2(3x+6)
(E) 6(x + 2)
Answers
3(2x+4), 2(3x+6) and 6(x + 2) are the expressions are equivalent to 6X +12
What is Expression?An expression is combination of variables, numbers and operators.
The given expression is 6X +12
We have to find the equivalent expressions of 6x+12
Let us take 6 as a common factor we get
6(x+2)
so 6(x+2) is equivalent expression of 6X +12
Now let us take 3 as common factor
3(2x+4)
so 3(2x+4) is equivalent expression of 6X +12
Now let us take 2 as common factor
2(3x+6)
so 2(3x+6) is equivalent expression of 6X +12
Hence, 3(2x+4), 2(3x+6) and 6(x + 2) are the expressions are equivalent to 6X +12
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