Answers
Given that y₁(x) = x is a solution to the given second-order ODE, assume a second solution of the form
y₂(x) = y₁(x) v(x) = x v(x)
Take the first two derivatives and substitute this solution into the ODE:
y₂ = x v
y₂' = x v' + v
y₂'' = (x v'' + v' ) + v' = x v'' + 2 v'
→ (1 - x ²) y₂'' - 2 x y₂' + 2 y₂ = 0
→ (1 - x ²) (x v'' + 2 v' ) - 2 x (x v' + v) + 2 x v = 0
→ (x - x ³) v'' + (2 - 4 x ²) v' = 0
Next, reduce the order by substituting w(x) = v'(x) and w'(x) = v''(x) :
→ (x - x ³) w' + (2 - 4 x ²) w = 0
and now you have a linear equation in w. Solve for w(x) however you like; I'll just separate variables.
Write w'(x) = dw/dx, then separate variables as
(x - x ³) dw/dx + (2 - 4 x ²) w = 0
(x - x ³) dw/dx = (4x ² - 2) w
dw/w = (4x ² - 2) / (x - x ³) dx
Integrate both sides:
∫ dw/w = (4x ² - 2) / (x - x ³) dx
ln|w| = ∫ (4x ² - 2) / (x - x ³) dx
For the remaining integral, expand the integrand into partial fractions.
x - x ³ = x (1 - x ²) = x (1 - x) (1 + x)
→ (4x ² - 2) / (x - x ³) = a /x + b / (1 - x) + c / (1 + x)
→ 4x ² - 2 = a (1 - x ²) + b x (1 + x) + c x (1 - x)
→ 4x ² - 2 = (-a + b - c) x ² + (b + c) x + a
→ -a + b - c = 4, b + c = 0, a = -2
→ a = -2, b = 1, c = -1
→ (4x ² - 2) / (x - x ³) = -2/x + 1 / (1 - x) - 1 / (1 + x)
Then
ln|w| = ∫ (-2/x + 1 / (1 - x) - 1 / (1 + x)) dx
ln|w| = -2 ln|x| - ln|1 - x| - ln|1 + x| + C
ln|w| = -ln|x ² (1 - x ²)| + C
exp(ln|w|) = exp(-ln|x ² (1 - x ²)| + C )
w = C / (x ² (1 - x ²))
v' = C / (x ² (1 - x ²))
Solve for v(x) by integrating both sides, expanding the right side into partial fractions again:
x ² (1 - x ²) = x ² (1 - x) (1 + x)
→ 1 / (x ² (1 - x ²)) = a / x + b / x ² + c / (1 - x) + d / (1 + x)
→ 1 = a x (1 - x ²) + b (1 - x ²) + c x ² (1 + x) + d x ² (1 - x)
→ 1 = (-a + c - d) x ³ + (-b + c + d) x ² + a x + b
→ -a + c - d = 0, -b + c + d = 0, a = 0, b = 1
→ a = 0, b = 1, c = 1/2, d = 1/2
→ 1 / (x ² (1 - x ²)) = 1/x ² + 1/2 (1 / (1 - x) + 1 / (1 + x))
dv/dx = C (1/x ² + 1/2 (1 / (1 - x) + 1 / (1 + x)))
dv = C (1/x ² + 1/2 (1 / (1 - x) + 1 / (1 + x))) dx
Integrate both sides:
∫ dv = ∫ C (1 / x ² + 1/2 (1 / (1 - x) + 1 / (1 + x))) dx
v = C₁ (-1/x - 1/2 ln|1 - x| + 1/2 ln|1 + x|) + C₂
v = C₁ (1/2 ln|(1 + x) / (1 - x)| - 1/x) + C₂
Solve for y₂(x) by multiplying both sides by x :
y₂ = C₁ (1/2 x ln|(1 + x) / (1 - x)| - 1) + C₂ x
Now, y₁(x) = x, so we can omit the last term from the second solution, giving us
y₂(x) = C (1/2 x ln|(1 + x) / (1 - x)| - 1)
Related Questions
If you know this you’re very smart!
Answers
Answer:
Its B and its not that hard because im doing that rn lol but do you need me to explain?
Step-by-step explanation:
63°
13°
Choose the equation that best describes the image.
Answers
Answer:
13+63 if I'm not wrong......
In the formula, l = a + (n - 1)d make d as the subject. Find d when l = 10, a = 2, n = 5.
Answers
Answer:
d=2
Step-by-step explanation:
l=a+(n-1) d
10=2+(5-1)d
10-2=4d
8=4d
d=2...
Answer:
2
Step-by-step explanation:
I = a + (n - 1)d
10 = 2 + (5 - 1)d
10 = 2 + 4d
10 - 2 = 4d
8 = 4d
d = 2
5th grade math. correct answer will be marked brainliest
Answers
Answer:
2 over 15 I think. I might be wrong. I rarely deal with fractions anymore.
25 points!!!
A system of equations is shown.
2x + 2y = 24
X = 3y
What is the value of x + y?
Enter your response here: Only 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -, -, and / are allowed in your answer.
Answers that are mixed numbers must be entered as an improper fraction or a decimal.
Answers
Answer:
x+y=12
Step-by-step explanation:
First, substitute 3y for the spots with x:
2(3y) + 2y = 24
6y+ 2y = 24
8y= 24
y= 3
Then sub in 3 in the y place:
x= 3(3)
x= 9
Then plug in the x and y values in the equation:
9 + 3 =12
Good Luck!!!
plz mark me Brainliest!
Are these triangles congruent ?
Answers
Answer:no
Step-by-step explanation:
Find the critical value t for the confidence level c= 0.98 and sample size n=7.
Click the icon to view the t-distribution table.
=(Round to the nearest thousandth as needed.)
Answers
Answer:
The critical value is T = 3.143.
Step-by-step explanation:
The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So
df = 7 - 1 = 6
98% confidence interval
Now, we have to find a value of T, which is found looking at the t table, with 6 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.98}{2} = 0.99[/tex]. So we have T = 3.143.
The critical value is T = 3.143.
Manny goes bowling.
- He has $25.00 to spend.
- He spends $4.25 to rent shoes.
- He spends $2.50 for each game he bowls.
Which inequality can Manny use to determine x, the greatest number of games he can bowl?
A. 2.5+4.25x≥25
B.4.25+2.5x≥25
C. 2.5+4.25x≤25
D. 4.25+2.5x≤25
Answers
Answer:
D
Step-by-step explanation:
Four equal sized boxes contain a
maximum of 80 books. How many
books will fit in 10 boxes?
Answers
If one box contains 20 books, then 10 boxes should fit 10 • 20 = 200 books.
someone please help me answer this!!
Answers
Answer:
4 1/10
Step-by-step explanation:
4.1
= 4 + 1/10
= 4 1/10
Hope you understand!
The formula of calculating the gradient of a line is
Answers
Answer:
To calculate the gradient of a straight line we choose two points on the line itself. From these two points we calculate: The difference in height (y co-ordinates) ÷ The difference in width (x co-ordinates). If the answer is a positive value then the line is uphill in direction.
a cone has a height of 35 and a radius of 19
Answers
Answer:
Step-by-step explanation:
The radius should be 17.5 if the height is 35. And also what is the question?
Answer:
Base area = 1,134.115
Lateral surface = 2,377.142
Slant height = 39.825
Surface area = 3,511.257
Volume = 13,231.341
Step-by-step explanation:
In a class of 50 students, 20 play hockey and 35 play football . 10 students play both hockey and football.
Answers
What is the median value of the data set shown on the line plot?
Enter your answer in the box.
Help
Answers
Answer: 30
Step-by-step explanation:
You put the numbers from least to greatest. 27,28,28,29,29,29,30,30,30,31,31,31,33,34. Then the median would be 30 and 30. So then you would add the two numbers 30 + 30 then divide by 2. 60/2 and the answer would be 30. Hopefully that was helpful.
exponential 2^2/2^-3
Answers
Answer:
32
Step-by-step explanation:
2 ^(2- -3)
= 2^5
Answer:
32
Step-by-step explanation:
[tex] \dfrac{2^2}{2^{-3}} = 2^{2 - (-3)} = 2^{2 + 3} = 2^5 = 32 [/tex]
If the figure is reflected over the x-axis, what are the coordinates of point R?
Answers
Answer:
R' (3,-3)
Step-by-step explanation:
Kenzie finds purple, blue, and silver space rocks while exploring, 2/3 of the rocks are purple and 3/4 of the remainder are blue. If there are 3 silver rocks, how many space rocks does Kenzie have?a
Answers
Answer:
There were 36 space rocks in total.
Step-by-step explanation:
First, you need to find what fraction of the original rocks are silver.
If 2/3 are purple, then 1/3 of them are not.
If 3/4 of the remaining 1/3 are blue, then 3/4 x 1/3 = 3/12 rocks are blue.
2/3 of the rock are purple, 3/12 of them are blue, and the remaining is silver.
So, you need to do 2/3 - 3/12, then find common denominators so, 8/12 - 3/12 which = 1/12.
This means 1/12 of the rocks are silver. If there are silver rocks, you need to find a suitable equation that needs to be solved.
So:
1/12r = 3
r = 36 silver rocks
There were 36 space rocks in total.
hope this helps :)
Plz help! Urgent!
Use the information in the table and choose all true statements.
Answers
Answer:
Pretty sure it's the first 3 options.
Step-by-step explanation:
Hi first of all, im totally new at this, but this seemed like a simple q. If anyone sees this and finds it wrong, pls pls let me know!
China and USA have highest/lowest density. You can tell by looking at the chart.
Indias density is about 360 people/km^2
Divide India's population by the Area
= 360.22
And the last option, according to me, is wrong.
Hope it helps !! And if it doesnt, again, pls let me know :)
Find the total surface area. Round to the nearest hundredth if necessary.
Answers
Answer:
166
Step-by-step explanation:
multiply the 6 by 2 and then multiply 13 by 10
to get 166
10 x __= 5/6 + 5/6 Solve.
Answers
Answer:
[tex]let \: that \: number \: be \: m \\ 10 \times m = \frac{5}{6} + \frac{5}{6} \\ 10m = \frac{10}{6} \\ m = \frac{10}{60} \\ m = \frac{1}{6} [/tex]
What’s the circumference?
Use 3.14 for pi
Extra large pizza: 20in diameter
Large pizza: 16in diameter
Medium pizza: 12in diameter
Small pizza: 10in diameter
If you help me with all of them I’ll give BRAINLIEST!!!
Answers
Answer:
all the stuff written there are answers if I'm wrong tell me I'll try it out again have a good day.
Which value is equivalent to V-25?
A. -5i
B. -5
C. 5
OD. 5i
Answers
Answer:
[tex]\sqrt{-25} = 5i[/tex]
Step-by-step explanation:
Given
[tex]\sqrt{-25[/tex]
Required
An equivalent
We have:
[tex]\sqrt{-25[/tex]
Express -25 as -1 * 25
[tex]\sqrt{-25} = \sqrt{-1 * 25}[/tex]
Split
[tex]\sqrt{-25} = \sqrt{-1} * \sqrt{25}[/tex]
[tex]\sqrt{25} = 5[/tex], so:
[tex]\sqrt{-25} = \sqrt{-1} * 5[/tex]
[tex]\sqrt{-1} = i[/tex] --- complex numbers
So:
[tex]\sqrt{-25} = i * 5[/tex]
[tex]\sqrt{-25} = 5i[/tex]
Hence, the equivalent is (D)
What is the slope of the line perpendicular to 2x-3y=8
Answers
Answer:
y = - [tex]\frac{3}{2}[/tex]x - [tex]\frac{8}{3}[/tex]
Step-by-step explanation:
2x-3y=8
rewrite, solve for y
3y = 2x - 8
divide by 3
y = [tex]\frac{2}{3}[/tex]x - [tex]\frac{8}{3}[/tex]
perpendicular = negative reciprocal slope = - [tex]\frac{3}{2}[/tex]
Can someone help me with 1 2 and 3 please
Answers
Answer:
1. vertical angles
2. 90°, complementary
3. 180°, supplementary
Answer:
1) Vertical angles
2) 90°, complementary
3) 180°, supplementary
Step-by-step explanation:
1) Intersecting lines form vertical angles.
According to the word problem, ∠APC and ∠DQB are formed by intersecting lines, meaning they are vertical angles.
2) ∠AQV is 30°, and ∠VQT is 60°.
30° + 60° = 90°. The sum of these angles is 90°.
And, complementary angles are two angles with measures that sum up to 90°.
3) ∠AQC is 150°, because 30° + 60° + 60° = 150°.
∠CQB is 30°.
And, 150° + 30° = 180°.
Supplementary angles are two angles with measures that sum up to 180°.
25, 87, 90, 25, 67
Mean:
Median:
Mode:
Range:
500,100,300,300,500,100
Mean:
Median:
Mode:
Range:
5,5,4,4,3,3,2
Mean:
Median:
Mode:
Range:
Answers
Answer:
mean number 1 25+87+90+25+67=294 294/5 58.8 question 2 500+100+300+300+500+100=1800 1800/6 =300
Answer:
Step-by-step explanation:
Mean: 289 (not sure)
Median: 90
Mode: 25
Range: 65
Mean: 1794 (still not sure)
Median: 300
Mode: 400?
Range: 400
Mean: 19
Median: 4
Mode: i dont know
Range: 3
The vertices of ABC are A(-2,4), B (-2,2), and C (-5,2). If ABC is reflected across the line y=1 to produce the image A”B”C find the coordinates of the vertex B’
Answers
The vertices of ∆ABC are A (-2,5), B(-2,4), and C(-5, 3). If ∆ABC is reflected across the line y=-2 to produce the image ∆A’B’C, find the coordinates of the vertex A’ if ∆ABC is reflected across the line x= -2.
State the name of the property illustrated. 5 + [7+(4)] = (5+7) + (4)
A. Communtative property of addition
B. Identity property of addiction
C. Associative property of addition
Answers
Which of these groups of relative frequencies would be best represented by a
pie chart?
A. 16%; 18%; 17%; 16%; 17%; 16%
B. 20%; 19%; 21%; 20%; 20%
C. 6%; 14%; 80%
D. 25%; 24%; 26%: 25%
Answers
Answer:
C
Step-by-step explanation:
different size wedges would be the most visibly different
Ricky is testing soil for a contaminant at a building site,
He'll take action to stop construction if there's strong
evidence that the soil has more than 400 parts per
million (ppm) of the contaminant. He plans on using soil
from n = 30 randomly selected locations at the building
site. His hypotheses are H, :p < 400 ppm and
H:u > 400 ppm, where u is the mean amount of the
contaminant in the soil at this site,
Suppose that in reality, H, is actually true,
Which situation below would result in the lowest
probability of a Type Il error?
Answers
The scenario that will bring about a type II error will be when the true mean is 405 ppm and he uses a significance level of 0.15
What is a type II error?It should be noted that a type II error simply describes the error that occurs when the individual fails to reject a null hypothesis.
From the complete question, type II error will be when the true mean is 405 ppm and he uses a significance level of 0.15. In this case, the null hyothesis was not rejected.
Learn more about type II error on:
https://brainly.com/question/16012410
Answer:
Step-by-step explanation:
Liam invests $5,000 in an account that pays 2.5% compound interest annually. He uses the expression P(1+r)t to find the total value of the account after t years. What will be the total value of the account after 3 years?
Answers
Answer:
The final balance is $5,389.
The total compound interest is $389.
Step-by-step explanation: