A Bernoulli differential equation is one of the formsdy/dx + P(x)y = Q(x)y^n (*) Observe that, if n 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y^(1-n) transforms the Bernoulli equation into the linear equation du/dx + (1 - n)P(x)u = (1 - n)Q(x). Consider the initial value problem xy’ + y = 2xy^2, y(1) = 4. (a) This differential equation can be written in the form (*) with P(x) = ___,Q(x) =___, and n = ___.(b) The substitution u = ___ will transform it into the linear equation Du/dx + ___ u = ___(c) Using the substitution in part (b), we rewrite the initial condition in part (c) u(1) = ____(d) Now solve the linear equation in part (b). and find the solution that satisfies the initial condition in part (c) U(x) = ____(e) Finally, solve for yy(x) = ____
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the answer for this question is i don't know
So , bye
(a) This differential equation can be written in the form (*) with P(x) = 1, Q(x) = 2x, and n = 2.
(b) The substitution u = 1/y will transform it into the linear equation Du/dx + 0u = -2x
(c)Using the substitution in part (b), we rewrite the initial condition in part u(1) = 1/y(1) = 1/4.
(d) the linear equation in part (b) is y = 1/(C - x) and the solution that satisfies the initial condition in part (c) U(x) = -y⁻¹ = -x + C,
(e) Finally, solve for y(x) = 1/(1/4 + 1 - x) = 4/(5 - x).
What is Bernoulli differential equation?
A Bernoulli differential equation is a type of non-linear ordinary differential equation of the form:
dy/dx + P(x)y = Q(x)yⁿ
where P(x) and Q(x) are functions of x and n is a constant. The equation is named after the Swiss mathematician Jakob Bernoulli.
These types of differential equations are important in various fields of science, such as physics, engineering, and economics. They are non-linear, meaning that the dependent variable y appears in both linear and non-linear terms, making them difficult to solve in general.
a) This differential equation can be written in the form (*) with P(x) = x, Q(x) = 2x, and n = 2.
b) The substitution u = y⁻¹ will transform it into the linear equation du/dx - xu = -2x.
c) Using the substitution in part (b), we rewrite the initial condition as u(1) = 1/4.
d) Now we solve the linear equation in part (b). The characteristic equation is given by d/dx(e\(\mathrm{(\int -xdx))}\) = -xe\(\mathrm{(\int -xdx))}\) which has the general solution u(x) = Ce\((-x^2/2)\).
Using the initial condition, we find C = 1/4, so u(x) = 1/4e\((-x^2/2)\).
e) Finally, we solve for y. From u = y⁻¹, we have y = 1/u = 4e\((-x^2/2)\).
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Related Questions
What is 906 times 45
Answers
Answer:
40770
........................
Answer:
40,770
Step-by-step explanation:
Determine the amplitude of function
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The amplitudes of functions are a) 8 and b) 6.
Given are the functions we need to determine the amplitude of function,
a) y = 8 Sin (x/2) + 3
b) y = 6 Cos x + 2
So,
To determine the amplitude of a trigonometric function, you can follow these steps:
For a sine function of the form y = A×sin(Bx + C) + D:
The amplitude is equal to the absolute value of the coefficient A.
For a cosine function of the form y = A×cos(Bx + C) + D:
The amplitude is equal to the absolute value of the coefficient A.
Let's apply these steps to the given functions:
a) y = 8×sin(x/2) + 3
The coefficient of sin in this function is 8, so the amplitude is |8| = 8.
Therefore, the amplitude of function a) is 8.
b) y = 6×cos(x) + 2
The coefficient of cos in this function is 6, so the amplitude is |6| = 6.
Therefore, the amplitude of function b) is 6.
Hence the amplitudes of functions are a) 8 and b) 6.
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What is the solution of the linear equation? LaTeX: 5k\:+\:3.8\:=\:3k\:+\:95 k + 3.8 = 3 k + 9 Group of answer choices 26 6.4 .065 2.6
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Answer:
\(k = 2.6\)
Step-by-step explanation:
Given
\(5k + 3.8 = 3k + 9\)
Required
Solve
\(5k + 3.8 = 3k + 9\)
Collect like terms
\(5k -3k+ 3.8 = 3k -3k + 9\)
\(2k+ 3.8 = 9\)
Subtract 3.8 from both sides
\(2k+ 3.8 - 3.8= 9 - 3.8\)
\(2k= 9 - 3.8\)
\(2k = 5.2\)
Divide through by 2
\(k = 5.2/2\)
\(k = 2.6\)
Given a sphere with a diameter of 6.2cm, find its volume to the nearest whole
Answers
Answer:
the answer is 12
Step-by-step explanation:
d=2r=2*6.2=12.4
4. Mrs. Magee's class is collecting books to donate to the local library. The
class goal is to collect 60 books for the youth section, and Mrs. Magee's
student's have donated 15 books already. They plan to donate 3 more books
every week until they reach their goal. Which of the following could be used
to determine w, the number of additional weeks that Mrs. Magee's class
collects books?
Answers
cancerStep-by-step explanation:
(−1)^3⋅(−1)^2
simplify
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= -1
everything is on the photo
Answers
The triangle is an isosceles right triangle, thus, the correct option is B.
Which type of triangle is in the image?On the image we can see a right triangle (we know that for the square angle in the bottom right side), we can see that the two catheti of the right triangle have a small segment on the middle of them.
That notation is used to whos that the two sides have the same measure.
So we have a right triangle with two equal sides, so this is an isosceles right triangle.
The correct option is B.
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PLEASE HELP ME ON QUESTION ASAP!!
IF YOU HAVE A TOPIC LIST IN YOUR EXAMS AND IT SAYS AVERAGES AND THE RANGE ARE YOU GOING TO BE HAVING MEAN AND RANGE IN YOUR TEST OR MEAN, RANGE MODE, MIDPOINT BASICALLY ALL OF IT ? IF ANSWERS CORRECT ILL RATE YOU FIVE STARS, GIVE YOU A THANKS AND MAYBE EVEN BRAINLIEST (sorry for caps)
Answers
Answer:
Step-by-step explanation:
Typically yes you need to know
Mean
Median
Mode
and Range
Mean = average, add all numbers then divide by how many
Median = midpoint, middle number. Be sure to list numbers from small to large if there are 2 middle numbers (this happens when there are an even amount), take the average of the 2 middle numbers
Mode = numbers that occurs the most in the list of numbers
Range = This is the largest number minus the smallest number.
what are two ratios that are equivalent to 12:13?
Answers
Answer:
24:26 and 36:39 are two ratios that are equivalent to 12:13.
Step-by-step explanation:
Which is the correct answer
Answers
Answer:
Expotential Growth
Based only on the information given in the diagram, which congruence theorems or postulates could be given as reasons why ABC~XYZ? Check all that apply
Answers
The congruence theorems or postulates that are taken to be reasons why ABV≈ XYZ are SSS and SAS. That is option A and C.
What is the congruence theorem of similar triangles?The congruence theorem of triangle consist of various theorem that shows how two triangles are similar. They include the following:
Side side side theorem(SSS): This states that triangles are congruent if the three sides of one are equal to the three sides of another one.
Side angle side theorem(SAS): This states that triangles are congruent if two sides and one angle (between the sides) of one triangle are equal to two sides and one angle of another triangle.
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James is trying to save money for college. When
he is 5 years old, his uncle puts $750 into a
savings account for him. The money should
increase by 5% each year until he needs it for
college.
Write an exponential model for this situation.
Answers
Answer:
\(y=(750)1.05^t\), where \(t\) is the time (in years) from investment
Or \(y=(750)1.05^{t-5}\), where \(t\) is the age of James
Step-by-step explanation:
exponential model: \(y=ab^t\), where \(t\) is the time (in years) from investment
when t = 0, y = 750
\(\implies 750=ab^0\)
As \(b^0=1\)
\(\implies 750=a\)
Therefore, \(y=750b^t\)
If the savings increase by 5% each year, then b = 105%.
105% = 105/100 = 1.05
Therefore, b = 1.05
So final model: \(y=(750)1.05^t\), where \(t\) is the time (in years) from investment
Or, we can write this as \(y=(750)1.05^{t-5}\), where \(t\) is the age of James
The population of a country has been increasing for several decades. In 1990, the population was about 263 million people In 2014, the population was about 307 million people Determine the percent in + The country's population increased by about % during this time period
Answers
Answer:
The country's population increased by about 16.73% during this time period.
Step-by-step explanation:
Percent increase:
The percent increase is given by the change multiplied by 100 and divided by the initial value.
In 1990, the population was about 263 million people In 2014, the population was about 307 million people.
Change: 307 - 263 = 44
Initial population: 263
Percentage increase:
44*100/263 = 16.73%
The country's population increased by about 16.73% during this time period.
I need help on this question
Answers
Answer:
(-2,1)
Step-by-step explanation:
We want the points where the function is greater than zero
Looking at the function it is between the x intercepts
The x intercepts ares -2 and 1
-2 <x<1 is when y >0
We want interval notation
(-2,1)
x and y are both differentiable functions of t x^2 + 3xy − y^2=9
Find dy/dt when x = 2 given dx/dt
= −1
Answers
The value of dy/dt is 4/3.
We have,
x² + 3xy - y² = 9
Now differentiating above equation w r t to 't' we get
d/dt x² + d/dt (3xy) - d/dt (-y²) = 0
d/dx (x²) . dx/dt + 3 dx/dt. dy/dt - 2y dy dt = 0
2x (-1) + 3(-1) dy/dt - 2y dy/dt = 0
dy/dt(-3 -2y)= -4
dy/dt = -4/(-3)
dy/dt = 4/3
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3 • f (-4) -3 • g (-2)=
Answers
Answer:
F=g^2+6
Step-by-step explanation:
Simplify:
1. Assume you need to multiply the variables by the numbers next to them:
\(3 * f (-4) - 3 * g (-2)=-12f + 6g\)2. Assume the number next to the variables mean exponents:
\(3 * f ^{-4} - 3 * g^{-2}=3/f^4-3/g^2=3(g^2-f^4)/f^4g^2\)which expression is equivalent to –3 – 3x – 1 + x 2x – 4 –2x + 4 –2x – 4? 4 – 2x
Answers
Answer:
you didn't give any options, I could simplify it... but that doesn't show which expression is equivalent.
Step-by-step explanation:
Answer:
–2x – 4?
Step-by-step explanation:
what is
(2ab-a+2) - (-ab-2)
Answers
Step-by-step explanation:
iknow but its hard to expalin
for each parallel lines. you are given the measure of one angle
Answers
Answer:
The question is not complete
Calculate the missing values in these similar triangles
Please show work , if u can.
Thank you <3
Answers
Answer:
x = 16
y = 70.4
Step-by-step explanation:
Two triangles are similar, so we can use the similarity ratio to so find the variables
(2.8)/(22.4) = (8.8)/y cross multiply expressions
197.12 = (2.8)*y divide both sides by 2.8
70.4 = y
use same operation to find x
(2.8)/(22.4) = x/128 cross multiply expressions
358.4 = (22.4)*x divide both sides by 22.4
16 = x
A cylinder has a height of 5 yards and a radius of 2 yards. What is its volume? Use ≈ 3.14 and round your answer to the nearest hundredth.
Answers
The volume of the cylinder is approximately 62.83 cubic yards.
The volume of a cylinder is given by:
V = πr²h
r is the radius of the cylinder h is the height of the cylinder and π is approximately equal to 3.14.
Substituting the given values we get:
V = 3.14 × 2² × 5
Simplifying we get:
V = 3.14 × 4 × 5
= 62.8
Rounding to the nearest hundredth we get:
V ≈ 62.83
A cylinder's volume is determined by:
V = r2h r is the cylinder's radius The cylinder's height is h and the value of is around 3.14.
By substituting the provided values, we obtain:
V = 3.14 × 2² × 5
If we simplify, we get:
V = 3.14 × 4 × 5 = 62.8
To the closest hundredth, we round to:
V ≈ 62.83
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Which sum or difference is modeled by the algebra tiles?
Answers
As a result, choice C is the equation that best represents the algebraic tiles since we only need to determine the total or difference.
what is expression ?It is possible either multiply, distribute, add, or negate in mathematics. The trying to follow is how a expression is put together: Number, expression, and scientific operator The ingredients of a mathematical expression include numbers, variables, and operations (such as addition, subtraction, multiplication or division etc.) It is possible to combine expressions and phrases. An expressions, often known as an arithmetic operation, is any mathematical statement that contains variables, numerals, and a numerical operation between them. That instance, the word m there in given equation is separated from the terms 4m and 5 by the arithmetic symbol +, as is the variable m in the phrase 4m + 5.
given
To address this issue, we just need to determine the total or difference that yields 2x2 + 2x + 2
Option A: ( x2 + 4x -2) + (x2 - 2x - 4) = 2x2 + 2x - 6
Option B: ( x2 + 4x - 2 ) + ( x2 + 2x + 4 ) = 2x2 + 6x + 2
Option C: ( x2 + 4x - 2) - ( -x2 + 2x - 4)= x2 + 4x - 2 + x2 - 2x + 4
= 2x2 + 2x + 2
As a result, choice C is the equation that best represents the algebraic tiles since we only need to determine the total or difference.
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The complete question is:-
Select the correct answer.
Which sum or difference is modeled by the algebra tiles?
A. (-x2 + 2x + 3) − (x2 − 2x − 1) = -2x2 + 2
B. (-x2 + 2x + 3) − (-x2 + 2x + 1) = -2x2 + 2
C. (-x2 + 2x + 3) + (-x2 + 2x + 1) = -2x2 + 2
D. (-x2 + 2x + 3) + (-x2 − 2x − 1) = -2x2 + 2
10. Find \(\frac{d^{2}}{d x^{2}}\int ^x_0\left(\int ^{\sin t}_1\sqrt{1+u^{4}}du\right)dt\text{.}\)
Answers
Let g(t) denote the inner integral. By the fundamental theorem of calculus, the first derivative is
\(\displaystyle \frac{d}{dx} \int_0^x g(t) \, dt = g(x)\)
Then using the FTC again, differentiating g gives
\(\displaystyle \frac{dg}{dx} = \frac{d}{dx} \int_1^{\sin(x)} \sqrt{1+u^4} \, du = \boxed{\cos(x) \sqrt{1+\sin^4(x)}}\)
A quadratic function y=f(x) is plotted on a graph in the vertex of the resulting parabola is (5,-5). what is the vertex of the function defined as g(x) = f(x+3)
Answers
9514 1404 393
Answer:
(2, -5)
Step-by-step explanation:
Using x+3 in place of x causes the function graph to be shifted left 3 units. That is, 3 is subtracted from each x-coordinate.
The vertex coordinates (5, -5) get translated to (5-3, -5) = (2, -5).
24 children were asked which is their favourite colour from a choice of red, blue
green or yellow. Their answers are shown below:
Blue
Red
Red
Green
Yellow
Red
Red
Red
Complete the frequency table.
Colour
Red
Blue
Red
Yellow
Red
Blue
Green
| Yellow 1111
Green
Blue
Green
Blue
Tally
Un Un
un
Analysing and Displaying Data
Yellow
Yellow
Blue
Red
Frequency
Red
Red
Blue
Yellow
Answers
The frequencies will be Red : 6, Blue: 4, Green: 8, Yellow : 1 (half of 2), Pink: 1 (half of 2).
What is a conditional relative frequency?It is defined as the frequency that can be evaluated by the two-way frequency table. Usually, we can obtain this frequency by dividing the frequency that is not in the total frequency cell to total frequency.
It is given that:
24 children were asked which is their favourite colour from a choice of red, blue
green or yellow
From the data given in the question.
Red : 6
Blue: 4
Green: 8
Yellow : 1 (half of 2)
Pink: 1 (half of 2)
Thus, the frequencies will be Red : 6, Blue: 4, Green: 8, Yellow : 1 (half of 2), Pink: 1 (half of 2).
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A spherical chocolate covered candy hasa diameter of 8cm. 4Part of the candy is a chocolate shell thatis .5cm thick.What is the volume ofjust the chocolate shell?
Answers
We want to calculate the volume of the spherical shell of chocolate.
The chocolate shell is spherical with a thickness of 0.5cm
Volumes of spherical shells can be calculated with the formula;
\(V_{shell}=\frac{4}{3}\pi(R^3-r^3)\)The outer diameter is 8cm. The outer radius is therefore 4cm
The inner diameter is 8-2(0.5)=7cm. The inner radius is therefore 3.5cm
Therefore, we can find the volume of the chocolate shell as;
\(V_{shell}=\frac{4}{3}\pi(4^3-3.5^3)=88.5\operatorname{cm}^3\)Therefore,
\(V_{shell}=88.5\operatorname{cm}^3\)Given the formula A = 5h (B + b); solve for B.
2
Answers
Answer:
A=5h(B+b)
A/5h=B+b
A/5h - b= B
The population of a city has increased by 35% since it was last measured. If the current population is 29,700 , what was the previous population?
Answers
Answer:
19305
Step-by-step explanation:
We simply take the percentage of 29700 to find how many people were added.
29700(0.35) = 10395 <== so 10395 people have been added
Subtract it from the current:
28700 - 10395 = 19305 people before.
The drawing shows a semicircular window separated into 3 sections of . Red Green Green Approximately how many of the glass are red? The window a diameter of 18 units
Answers
Answer:
\( \frac{45π}{2} \: units^{2} \: ≈ \: \boxed{70.7 \: units²} \)
______________________
The correct option is 70.7
Step-by-step explanation:
Since the green and red make a semicircle, the total measure must be 180°.
Since the two green sectors are both 40°.
The red must add up with the green sectors to total 180°.
This means that:
red sector + 80° = 180°.
–80° –80.
red sector = 100°.
To find the area of a sector, we must first understand the area of the circle itself which is πr², where r is the radius.
Since a full circle is 360°.
(360° / 360°)( πr² ) will be the area.
Or in radians: (2π rad / 2π rad)( πr² ).
From here, we can create the formula:
Area of a sector = ( n° / 360° ) ( πr² ). Where n is the measure or the sector in degrees, and r is the radius.
You may also know that the diameter is twice the measure of the radius.
This means that if we are given a diameter of 18 units from the problem, the radius will be 18/2 or 9 units.
Lastly, all we have to do is substitute all this information to find the area of sector.
Area of the red sector = ( n° / 360° ) ( πr² ).
Area of the red sector =
( (100°) / 360° ) ( π(9)² ).
Area of the red sector = ( 5 / 18 ) ( 81π )
Area of the red sector = ( (5)(81π) / 18 )
Area of the red sector = ( (405π) / 18 )
Area of the red sector = ( 45π / 2 ) units²
Area of the red sector =
(141.371669412.. units²) / 2
Area of the red sector =
70.6858347058.. units²
Area of the red sector ≈
70.7 units²
Find the area of the triangle.
7 yd
24 yd
The area is
square yards.
Answers
Answer:
84 yd^2
Step-by-step explanation:
Answer:
½ × 7yd × 24yd = 84yd² <<<<< the answer