find the differential equations
x2y" —2y' + xy= 0 (x2-9)y" - 2xy + y= 0
Answers
The given differential equations are: x^2y" - 2y' + xy = 0 (x^2 - 9)y" - 2xy + y = 0, These are second-order linear homogeneous differential equations. To solve them we would use substitution and simplification.
x^2y" - 2y' + xy = 0:
To solve this equation, we can assume a solution of the form y = x^r, where r is a constant. Taking the derivatives, we get:
y' = rx^(r-1)
y" = r(r-1)x^(r-2)
Substituting these derivatives into the differential equation, we have:
x^2(r(r-1)x^(r-2)) - 2(rx^(r-1)) + x(x^r) = 0
Simplifying, we get:
r(r-1)x^r - 2rx^r + x^(r+1) = 0
Now, divide through by x^r to eliminate the x term:
r(r-1) - 2r + x = 0
This equation should hold for all x, so the coefficient of x must be zero. Therefore:
r(r-1) - 2r = 0
Simplifying, we have:
r^2 - 3r = 0
Factoring out r, we get:
r(r-3) = 0
So, the solutions for r are r = 0 and r = 3.
Therefore, the general solution of the differential equation is:
y = c1x^0 + c2x^3
Simplifying further, we have:
y = c1 + c2x^3
where c1 and c2 are constants.
(x^2 - 9)y" - 2xy + y = 0:
This equation can be simplified by factoring out (x^2 - 9), which is (x - 3)(x + 3):
(x - 3)(x + 3)y" - 2xy + y = 0
Now, we can divide through by (x - 3)(x + 3) to get:
y" - (2x / (x - 3)(x + 3))y + (1 / (x - 3)(x + 3))y = 0
This equation is a Cauchy-Euler equation, which can be solved by assuming a solution of the form y = x^r:
r(r-1)x^(r-2) - (2x / (x - 3)(x + 3))rx^(r-1) + (1 / (x - 3)(x + 3))x^r = 0
Multiplying through by (x - 3)(x + 3)x^(2-r), we get:
r(r-1)x^2 - 2rx^2 + x^2 = 0
Simplifying, we have:
r(r-1) - 2r + 1 = 0
(r^2 - 2r + 1) = 0
(r-1)^2 = 0
So, the solution for r is r = 1.
Therefore, the general solution of the differential equation is:
y = c1x^1 + c2x^1 ln|x|
Simplifying further, we have:
y = c1x + c2x ln|x|
where c1 and c2 are constants.
These are the solutions to the given differential equations.
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Related Questions
Consider the following current information for Galaxy Inc::
Output = 200 units
ATC = $50
What is the total cost of producing 200 units of output?
a. $10,000
b. $8,000
c. $1,100
d. Non
Answers
The answer is (a) $10,000.
How the total cost of producing 200 units of output can be found?The total cost of producing 200 units of output can be found by multiplying the output (200 units) by the average total cost (ATC) per unit, which is given as $50. Therefore, the total cost is:
Total Cost = Output x ATC
Total Cost = 200 x $50
Total Cost = $10,000
Therefore, the answer is (a) $10,000.
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Apply distributive property:
-6(2x+5)=
Answers
Answer:
-12x-30
Step-by-step explanation:
First you Distribute -6 times 2x= -12x
Then you multiply -6 times 5=-30
So it is -12x-30
Answer:
-12x-30
Step-by-step explanation:
-6*2x+(-6*5)=-12x-30
Find x:
(Round the answer to the nearest tenth if there is a decimal)
Answers
Answer:Angle x is congruent with the interior angle opposite side 8 (alternate interior angles)
Use tangent:
tan x = 8/15
x = arctan (8/15)
x = 28.1° (rounded)
Step-by-step explanation:
identify the values of a h and k y=5/x+6-2
Answers
The values of a, h and k on the function are given as follows:
a = 5.h = 6.k = -2.The meaning of the transformations is given as follows:
Multiplication by a = 5 -> vertical stretch by a factor of 5.h = 6 -> translation left 6 units.k = -2, translation down 2 units.What is a translation?A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.
The four translation rules for functions are defined as follows:
Translation left a units: f(x + a).Translation right a units: f(x - a).Translation up a units: f(x) + a.Translation down a units: f(x) - a.The parent function in this problem is given as follows:
y = 1/x.
Hence the meaning of the transformations is given as follows:
Multiplication by a = 5 -> vertical stretch by a factor of 5.h = 6 -> transformation f(x + 6), translation left 6 units.k = -2, transformation f(x) - 2, shift down 2 units.More can be learned about translations at brainly.com/question/28174785
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HURRY 15(-74a+51)=15(7a+456)
a.6
b.-5
c.-2
d.5
Answers
Answer:
-5 because of I'm right I am always right
Use the Integrating Factor Method to solve the following differential equations: x⁴ dy/dx + 2x⁴y = x⁴e⁻ˣ
a) Rewrite the equation in Standard Form. b) Identify P(x). c) Identify Q(x). d) Evaluate Integrating Factor. e) Solve for the general solution.
Answers
The equation in Standard Form. b) Identify P(x). c) Identify Q(x). d) Evaluate Integrating Factor. e) Solve for the general solution are given below:
a) Rewrite the equation in Standard Form:
To rewrite the equation in standard form, divide the entire equation by x⁴:
dy/dx + 2y = e^(-x)
b) Identify P(x):
In standard form, the coefficient of the y term is 2, which is the function P(x). So, P(x) = 2.
c) Identify Q(x):
In standard form, the right-hand side of the equation is e^(-x), which is the function Q(x). So, Q(x) = e^(-x).
d) Evaluate the Integrating Factor:
The integrating factor (IF) is given by the exponential of the integral of P(x) with respect to x. In this case, the integrating factor is:
IF = e^(∫P(x)dx) = e^(∫2dx) = e^(2x)
e) Solve for the general solution:
Multiply the entire equation by the integrating factor (IF = e^(2x)):
e^(2x) * (dy/dx + 2y) = e^(2x) * e^(-x)
Simplify the left side by applying the product rule of exponents:
(e^(2x) * dy/dx) + 2y * e^(2x) = e^(x)
Notice that the left side is now in the form (f(x)g(x))' = f'(x)g(x) + f(x)g'(x), where f(x) = y and g(x) = e^(2x). Apply the product rule and simplify further:
(d/dx)(y * e^(2x)) = e^(x)
Integrate both sides with respect to x:
∫(d/dx)(y * e^(2x)) dx = ∫e^(x) dx
Integrating the left side gives:
y * e^(2x) = ∫e^(x) dx = e^(x) + C₁, where C₁ is the constant of integration.
Finally, solve for y by dividing both sides by e^(2x):
y = (e^(x) + C₁) / e^(2x)
This is the general solution to the given differential equation using the Integrating Factor Method.
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4. Six workers are hired to weed a field by
hand. Each is given a plot which is 9x11
feet in size. What is the total area of the
field?
Answers
Answer:
594 square feet
Step-by-step explanation:
Area of a rectangle is length times width. We can find the area of each workers plot by multiplying 9 and 11 to get 99 sq feet. There is six workers, and therefore six plots of land, so we'd take our 99 sq feet and count that six times to get 594 square feet.
Answer:
594 square feet
Step-by-step explanation:
A teacher gives 5 points for correctly completing an "easy" homework question and 10 points for correctly completing a "challenge" question. To get an A on a homework assignment, a student must get at least 100 points. Using x to represent the number of easy questions and y the number of challenge questions, what inequality models the number of questions a student needs to complete correctly in order to earn an A on an assignment?
Answers
The inequality models the number of questions a student needs to complete correctly in order to earn an A on an assignment is x + y > 100
A teacher gives 5 points for correctly completing an "easy" homework question and 10 points for correctly completing a "challenge" question.
To get an A on a homework assignment, a student must get at least 100 points.
Let x to represent the number of easy questions and y the number of challenge questions,
x + y > 100
Therefore, the inequality models the number of questions a student needs to complete correctly in order to earn an A on an assignment is x + y > 100
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7: There are 13 visitors at Al Shaheed Park today. This is 20 % of the normal
amount of visitors. How many visitors would there normally be today?
Answers
Answer:
65
Step-by-step explanation:
20% is 1/5 of 100.
multiply 13 by 5 to get the original amount of numbers. you can proof check it by doing
13/65 which is 0.2. multiply the decimal by 100 and you get 20. which is 20%
anyone solve this pls :)
Answers
your ans is here........
What are the 5 rules of scientific notation?
Answers
5 rules of scientific notation which helps us to represent the numbers which are very huge or very tiny in a form of multiplication of single-digit numbers and 10 raised to the power of the respective exponent.
The exponent is either positive or negative depending on how big or little the number is. Learn about power and exponents to have a deeper understanding. The five golden rules of the scientific notation:
Base should be always 10The exponent must be a non-zero integer, that means it can be either positive or negativeThe absolute value of the coefficient is greater than or equal to 1 but it should be less than 10 Coefficients can be positive or negative numbers including whole and decimal numbersThe mantissa carries the rest of the significant digits of the number of the exponent.Hence there are the rules of the the scientific notation.
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Select the correct answer from each drop-down menu.
A certain distribution has a mean value of 100 and a standard distribution of 15,
Assuming the values are distributed normally, 95% of values will fall between
and
Answers
Answer:
70, 130
Step-by-step explanation:
Just took the Mastery test on PLATO and this answer was correct.
What is 871.816 divided by 7618.882 ?
Answers
Answer:
-0,8777 the answer is at least it seems
Step-by-step explanation:
0.1144283374 is its answer..
What is the product of (–0.5)(–1.2)(–5.22)? –9.8 –3.132 3.132 9.8What is the product of (–0.5)(–1.2)(–5.22)? –9.8 –3.132 3.132 9.8What is the product of (–0.5)(–1.2)(–5.22)? –9.8 –3.132 3.132 9.8What is the product of (–0.5)(–1.2)(–5.22)? –9.8 –3.132 3.132 9.8
Answers
Answer:
Step-by-step explanation:
1. (–0.5)(–1.2)(–5.22)= -3.132
Answer:
B
Step-by-step explanation:
How many feet is 48 inches
Answers
48/12=4 feet
Solve the following inequality, graph the solution, and write
the solution in interval notation.
3x + 7 ≤ 13x + 7 ≤ 1 AND 2x + 3≥ − 5
Answers
First, let's solve the inequality 3x + 7 ≤ 13x + 7 ≤ 1:3x + 7 ≤ 13x + 7 Subtract 13x from both sides. -10x + 7 ≤ 7 Subtract 7 from both sides. -10x ≤ 0
Divide both sides by -10, remembering to reverse the inequality .x ≥ 0. Now let's solve the inequality 2x + 3 ≥ -5:2x + 3 ≥ -5
Subtract 3 from both sides. 2x ≥ -8 Divide both sides by 2. x ≥ -4.Using both inequalities, the solution is x ≥ -4 and x ≥ 0. This is because both inequalities must be satisfied at the same time. In interval notation, the solution is (-∞, ∞), meaning that x can be any number from negative infinity to positive infinity.
This is because the solution includes all real numbers greater than or equal to -4 and all real numbers greater than or equal to 0.
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What is the polar form of Negative 9 StartRoot 3 EndRoot + 9 i?
Answers
Answer:
Step-by-step explanation:
In the rectangular complex number -9√3 + 9i, which has a standard form a + bi, the a = -9√3 and the b = 9. We need this in polar form (r, θ) where
\(r=\sqrt{a^2+b^2}\) and filling in:
\(r=\sqrt{(-9\sqrt{3})^2+(9)^2 }\) (notice we do not put the i in there with the 9).
\(r=\sqrt{243+81}\) so
r = 18. Now let's move on to the angle, which is a little more difficult. The angle is found in the inverse tangent ratio:
\(tan^{-1}(\frac{b}{a})\) so filling that in, we have:
\(tan^{-1}(\frac{9}{-9\sqrt{3} })=\frac{1}{-\sqrt{3} }\) Since tangent is the side opposite over the side adjacent, y is positive and x is negative in the second quadrant. This is a 30 degree angle in QII, which has a reference angle of 150 degrees. This angle in radians is \(\frac{5\pi}6}\), so the polar form of that number is (18, \(\frac{5\pi}{6}\))
Answer:
its d
Step-by-step explanation:
i guessed lol
How tall is the tower?X60°17 ft-50 ft[ ? ] ftRound to the nearest tenth.
![How tall is the tower?X6017 ft-50 ft[ ? ] ftRound to the nearest tenth.](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/3mEhuzGxyTiEZel8MkHVq0yXjgVENxmv.jpeg)
Answers
We can use the trigonometric tangent identity, this is:
\(\tan z=\frac{opposite\text{ side}}{\text{adjacent side}}\)Where z = 60°
opposite side = x
adjacente side = 50ft
Then:
\(\begin{gathered} \tan 60=\frac{x}{50} \\ x=50\times\tan 60 \\ x=50\sqrt[]{3}=86.6 \end{gathered}\)And the tower height is:
\(86.6+7=93.6\)Answer: 93.6 ft
Two 6-sided dice, one red and one green, are rolled. What is the probability that the red die shows an odd number and the green die shows a number that is a perfect square
Answers
Let us first identify the total number of possible outcomes. Since there are two 6-sided dice, there are 6 possible outcomes for each die.
Thus, the total number of possible outcomes is 6 x 6 = 36.To find the probability of the red die showing an odd number, we first need to identify how many odd numbers are on a 6-sided die. There are three odd numbers on a 6-sided die: 1, 3, and 5.
Therefore, the probability of the red die showing an odd number is 3/6 or 1/2.There is only one perfect square number on a 6-sided die: 4.
Therefore, the probability of the green die showing a perfect square number is 1/6.To find the probability of both events happening, we multiply the probabilities:1/2 x 1/6 = 1/12Therefore, the probability that the red die shows an odd number and the green die shows a number that is a perfect square is 1/12.
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\( \sqrt[14]{6.3} \)need help showing my son how to solve this problem. explanation is key.
![[tex] \sqrt[14]{6.3} [/tex]need help showing my son how to solve this problem. explanation is key.](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/nuHY62ZKttDWALoNx1Cnl0nMyoSt8hpt.jpeg)
Answers
Divide 6.3 meters of leather
between 14
Then
6.3 meters= 630 cm
Now divide
630/14 =
Find m,c,d (max common divisor), between 63 and 14. Its 7.
divide by 7, both numerator and denominator
90/2 = 45
Then answer is , each piece is 45 cm long
Susan drinks five-fourths of a quart of milk. Rewrite the sentence using a mixed number.
Answers
Answer:1/5?
Step-by-step explanation:
i think that is 4/5 x 1/4 so, 1/5?
American General offers a 9-year annuity with a guaranteed rate of 6.28% compounded annually. How much should you pay for one of these annuities if you want to receive payments of $1500 annually over the 9 year period? How much should a customer pay for this annuity? (Round to the nearest cent)
Answers
You should pay approximately $10,117.09 initially to secure the annuity and receive annual payments of $1500 over the 9-year period.
To find the cost of the annuity, we need to calculate the present value of the future payments. The present value represents the current worth of future cash flows, taking into account the interest earned or charged over time. In this case, we'll calculate the present value of the $1500 payments using compound interest.
The formula to calculate the present value of an annuity is:
PV = PMT × [1 - (1 + r)⁻ⁿ] / r
Where:
PV is the present value of the annuity (the amount you should pay initially)
PMT is the payment amount received annually ($1500 in this case)
r is the interest rate per period (6.28% or 0.0628)
n is the total number of periods (9 years)
Let's substitute the values into the formula:
PV = $1500 × [1 - (1 + 0.0628)⁻⁹] / 0.0628
Calculating this expression:
PV = $1500 × [1 - 1.0628⁻⁹] / 0.0628
PV = $1500 × [1 - 0.575255] / 0.0628
PV = $1500 × 0.424745 / 0.0628
PV ≈ $10117.09
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Find missing angleeeeeeeeeeeeeeeeeeeeeeee
Answers
Answer:
C) 80
Step-by-step explanation:
triangles always add up to 180 degrees, so 80+20=100 the missing angle is 80 degrees
Anya wanted a book that cost $34.99 regularly at the bookstore. If Anya had a coupon for 20% off any regularly priced item at the bookstore , what is the sale price of the book?
A) $14.99
B.$27.99
C.$32.99
D.$34.99
ILL GIVE BRAINLIEST FOR 2 OR 1 RESPONSE WITH AN EXPLANATION :’)
Answers
Answer:
$27.99
Step-by-step explanation:
34.99*.20 = 6.998
34.99-6.998 = 27.992
the answer rounded would be $27.99
Hope this helps
which of the following give chart elements a realistic, three-dimensional look?A. axisB. chart areaC. legendD. plot area
Answers
The option that gives chart elements a realistic, three-dimensional look is D. Plot Area.
The option that gives chart elements a realistic, three-dimensional look is the plot area. The plot area is the portion of the chart that displays the data points and is typically the main focus of the chart. Adding shading and depth to the plot area, can give the chart a more lifelike appearance and make the data more visually engaging. Additionally, three-dimensional charts can also be created by using specialized charting software that allows for the creation of 3D visualizations. These types of charts can be useful for highlighting patterns and trends in the data, especially when dealing with large datasets.
In a chart, the plot area is the space where the actual data points, bars, lines, or pie slices are displayed. By applying three-dimensional effects to the plot area, you can make the chart elements appear more realistic and visually appealing. This can enhance the viewer's understanding of the data being represented and make the chart more engaging.
In summary, to give chart elements a realistic, three-dimensional look, you should focus on the plot area (Option D). The axis (Option A), chart area (Option B), and legend (Option C) do not directly contribute to creating a three-dimensional appearance for the chart elements.
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slope:
y-intercept:
PLSPLSPLS HELP
Answers
Answer:
Slope: 1
Y-intercept: -1
Step-by-step explanation:
The line passes the y-axis at -1, the rise/run, which is used to calculate the slope is 2/2, and it equals to 1, so the slope is 1.
I hope it helps! Have a great day!
bren~
Round 1.029 to the nearest hundredth.
Answers
Answer:
1.03
Step-by-step explanation:
1.029 is closer to 1.03 than 1.02, so round it to 1.03.
Answer:
1.03
Step-by-step explanation:
Evaluate 3x² - 1 when x = 2.
A. 11
B. 35
C. 3
D. 5
Answers
Answer:
35
Step-by-step explanation:
All you need to do is substitute!
(3x2)^2-1
6^2-1
36-1
= 35
Hope this helps!!
RS has a midpoint at M(7, 10). Point R is at (8, 10). Find the coordinates of point s.
Answers
Answer:
S(6, 10)
Step-by-step explanation:
1. The Midpoint (in this case, M) will always be halfway between both R and S (or other characters in some cases).
2. As M is only one X value away from R, it is only 1 X value from S as well, but in the other direction.
3. (7, 10) - (1, 0) = (6, 10)
The (7, 10) is the Midpoint Coordinates.
The (1, 0) is the distance from M to R.
The (6, 10) is the coordinates of S.
Answer:
(6/10)
Step-by-step explanation:
4 - dictado audio you will hear a conversation. Listen carefully and write what you hear during the pauses. The entire conversation will then be repeated so you can check your work
Answers
According to the conversation we can infer that the information focused on the conversation of two people; one of them is studing for an exam.
What is the conversation about?To identify what is the conversation about we have to consider who is talking and what are the ideas that they share. In this case, we can infer that the main idea of this conversation is that one of them is studying for an exam this week.
Also, the second person is apologizing because he considered that he is distracting the other person. So, the correc information about de conversation is a short conversation about study.
Note: This question is incomplete. Here is the complete information:
Conversation:
Hello
Hello, How are you?
I am fine, and you?
i am fine too, What are you doing?
I am studing for an exam this week.
Oh, sorry I won´t distract you.
Don´t worry.
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