Find a power series solution to the differential equation at the point x0. (2+x2)y′′-xy′+4y=0
Answers
I'll assume you are looking for a series centered at [tex]x=0[/tex], which is an ordinary point for the ODE. Substituting
[tex]y=\displaystyle\sum_{n\ge0}a_nx^n[/tex]
into the ODE, we can rewrite it as
[tex]\displaystyle\sum_{n\ge2}n(n-1)a_nx^n+2\sum_{n\ge2}n(n-1)a_nx^{n-2}[/tex]
[tex]\,\,\,\,\,\,\,\,-\displaystyle\sum_{n\ge1}na_nx^n+4\sum_{n\ge0}a_nx^n=0[/tex]
[tex]\displaystyle\sum_{n\ge2}n(n-1)a_nx^n+2\sum_{n\ge0}(n+2)(n+1)a_{n+2}x^n[/tex]
[tex]\displaystyle\,\,\,\,\,\,\,\,-\sum_{n\ge1}na_nx^n+4\sum_{n\ge0}a_nx^n=0[/tex]
[tex]\displaystyle4(a_0+a_2)+3(a_1+4a_3)x+\sum_{n\ge2}\bigg[2(n+2)(n+1)a_{n+2}+(n^2-2n+4)a_n\bigg]x^n=0[/tex]
so the coefficients of the power series are defined by the recurrence
[tex]\begin{cases}a_0=a_0\\\\a_1=a_1\\\\a_n=-\dfrac{(n-3)^2+3}{2n(n-1)}a_{n-2}&\text{for }n\ge2\end{cases}[/tex]
For even [tex]n[/tex], i.e. [tex]n=2k[/tex] [tex](k\ge0)[/tex], we have
[tex]a_0=a_0[/tex]
[tex]a_2=-a_0[/tex]
[tex]a_4=-\dfrac4{4\times3\times2}a_0=\dfrac4{4!}a_0[/tex]
[tex]a_6=-\dfrac{12}{2\times6\times5}a_4=-\dfrac{12\times4}{2\times6!}a_0[/tex]
[tex]a_8=-\dfrac{28}{2\times8\times7}a_6=\dfrac{28\times12\times4}{2^2\times8!}a_0[/tex]
and so on. The pattern in the denominator is pretty clear, but we can also find a compact form for the numerator. When [tex]k=5[/tex], we can write it as
[tex]4\times12\times28\times52=4^4(1\times3\times7\times13)[/tex]
[tex]=4^4\bigg(1\times(1+2)\times(1+2+4)\times(1+2+4+6)\bigg)[/tex]
[tex]=4^4\bigg(1\times(1+2(1))\times(1+2(1+2))\times(1+2(1+2+3)\bigg)[/tex]
[tex]=4^4\displaystyle\prod_{i=1}^3\left(1+2\sum_{j=1}^{i-1}j\right)[/tex]
[tex]=4^4\displaystyle\prod_{i=1}^3(i^2-i+1)[/tex]
So in general we have
[tex]a_{2k}=\dfrac{(-1)^k2^k\displaystyle\prod_{i=1}^{k-1}(i^2-i+1)}{(2k)!}[/tex]
We can treat the odd-indexed terms similarly. For [tex]n=2k-1[/tex] [tex](k\ge1)[/tex] we have
[tex]a_1=a_1[/tex]
[tex]a_3=-\dfrac14a_1=-\dfrac3{2\times3\times2}a_1=-\dfrac3{2\times3!}a_1[/tex]
[tex]a_5=-\dfrac7{2\times5\times4}a_3=\dfrac{7\times3}{2^2\times5!}a_1[/tex]
[tex]a_7=-\dfrac{19}{2\times7\times6}a_5=-\dfrac{19\times7\times3}{2^3\times7!}a_1[/tex]
and so on. Again, the pattern in the denominator is simple. For [tex]k=5[/tex], we would get a numerator of
[tex]3\times7\times19\times39=3\times(3+4)\times(3+4+12)\times(3+4+12+20)[/tex]
[tex]=3\times(3+4(1))\times(3+4(1+3))\times(3+4(1+3+5))[/tex]
[tex]=\displaystyle\prod_{i=1}^4\left(3+4\sum_{j=1}^{i-1}(2j-1)\right)[/tex]
[tex]=\displaystyle\prod_{i=1}^4(4i^2-8i+7)[/tex]
and in general we'd have
[tex]a_{2k-1}=\dfrac{(-1)^{k+1}\displaystyle\prod_{i=1}^{k-1}(4i^2-8i+7)}{2^{k-1}(2k-1)!}[/tex]
Thus the power series solution to this ODE is
[tex]y(x)=\displaystyle\sum_{k\ge0}a_{2k}x^{2k}+\sum_{k\ge1}a_{2k-1}x^{2k-1}[/tex]
Attached below is a plot of a numerical solution (blue) compared to the first 9 terms [tex](0\le n\le8)[/tex] and first 21 terms [tex](0\le n\le21)[/tex] of the series solution over the interval [tex]|x|\le3[/tex], assuming initial values of [tex]y(0)=y'(0)=1[/tex] [tex](a_0=a_1=1)[/tex].
Related Questions
:what does this problem solve as
1/2(4x−5)+52=10
Answers
the temperature of a pot of water is shown. the temperature of the water changed -2.5 degree F per minute. What was the temperature after 20 minutes?
Answers
=-50
Atif has a z score of 1.02 from a converted iq score. from a z table, the percentage between his score and the mean is 34.61%. what percent of people have iqs at or lower than atif's score?
Answers
Explanation.
Refer to the diagram shown below.
When the z-score is z = 0, then the area under the distribution curve is 50%, which corresponds to the mean, μ.
The given z-score is greater than zero, so the shaded area is 34.61%. to the right of z=0
The area to the left of z = 1.02 is 50% + 34.61% = 84.61%.
This percentage represents the people with IQs at or lower than Atfif's score.
A condo in Culver City, California, listed for $1.4 million with 20% down and financing at 5% for 30 years. What's the monthly payment?
Answers
Answer:
Step-by-step explanation:
$6,014.40
(1 point) consider the function f(t)=⎧⎩⎨⎪⎪⎪⎪0,−5,−6,6,t<00≤t<11≤t<7t≥7;f(t)={0,t<0−5,0≤t<1−6,1≤t<76,t≥7; 1. write the function in terms of unit step function
Answers
Final answer:
The function f(t) can be represented in terms of the unit step function as f(t) = -5u(t) - u(t-1) + 6u(t-7), with each term turning on at specific breakpoints.
Explanation:
To write the function f(t) in terms of the unit step function, you can think of f(t) as a piecewise function that changes values at various breakpoints. The unit step function, denoted as u(t), is defined to be 0 when t < 0 and 1 when t ≥ 0. For the given function f(t), we can write it using the unit step function as follows:
f(t) = -5u(t) -u(t-1) + 6u(t-7)
This representation uses the properties of the unit step function to 'turn on' different function values at specified points. Each term corresponds to a segment of the piecewise function, with the coefficients representing the value of f(t) within each segment.
You are debating about whether to buy a new computer for $800.00 or a refurbished computer with the same equipment for $640.00. If you are paid $10.00/hour and your deductions are FICA (7.65%), federal tax withholding (12%), and state tax withholding (8%), how many hours do you have to work to pay for the new computer?
A) 111 hours
B) 89 hours
C) 98 hours
D) 87 hours
Answers
First calculate all the deduction
FICA = $10 per hour x 0.0765 = $0.77 per hr
Fed tax = $10 per hr x 0.12 = $1.2 per hr
State tax = $10 per hr x 0.08 = $0.8 per hr
Remaining pay = 10 – 0.77 – 1.2 – 0.8 = $7.23 per hour
Time required = $640 / $7.23 per hr
Time required = 89 hrs
After calculating deductions from the gross wage of $10.00, the net income per hour is $7.235. To buy the new computer for $800, we divide the cost by the net wage, resulting in approximately 110.58 hours. Therefore, the student would need to work about 111 hours to afford it.
To calculate the number of hours needed to work to pay for the new computer, we must first determine the net income per hour after all the deductions. With a gross wage of $10.00 per hour, the deductions would be:
FICA: 7.65% of $10.00 = $0.765Federal Tax: 12% of $10.00 = $1.20State Tax: 8% of $10.00 = $0.80This totals $2.765 in deductions per hour. Subtracting this from the gross wage gives a net income per hour:
$10.00 - $2.765 = $7.235.
To afford the new computer costing $800, we divide the cost of the computer by the net income per hour to find the total number of hours needed:
$800.00 / $7.235 ≈ 110.58 hours.
Since we cannot work a fraction of an hour, we will need to round up to the nearest whole number, which suggests that the student would need to work approximately 111 hours to afford the new computer.
Paul is riding his bike 48 miles to the beach. His average rate of speed is 13.8 miles per hour. Paul stopped for a snack after 3 hours. How many miles does he have to go after his snack? Round your answer to the nearest tenth if necessary.
Answers
13.8 miles per hour x 3 hours = 41.4 miles ( what he already rode)
48 - 41.4 = 6.6 more miles to go
50 POINTS!!!
Suzanne ran 3 3/4 miles in the morning and 4 1/2 miles in the afternoon. How many kilometers did she run altogether? Assume 1 mile = 1.6 kilometers.
Answers
3 3/4 = 15/4
15/4 + 18/4 = 8 1/4 or 33/4
33/4 * 1.6/1 = 52.8/4 = 13.2
When assuming 1 mile equals 1.6 kilometers, Suzanne ran 13.2 kilometers in all.
Answer:- Total distance traveled by Suzanne in kilometers=13.2 kilometers.
Explanation:-
Distance traveled by Suzanne in the morning=[tex]3\frac{3}{4}\text{ miles}[/tex]
Distance traveled by Suzanne in the evening=[tex]4\frac{1}{2}\text{ miles}[/tex]
Total distance traveled by Suzanne=[tex][3\frac{3}{4}+4\frac{1}{2}\text{ miles}[/tex]
[tex]=\frac{15}{4}+\frac{9}{2}\text{ miles}[/tex]
[tex]=\frac{15+18}{4}\text{ miles}[/tex]
[tex]=\frac{33}{4}\text{ miles}[/tex]
[tex]=8.25\text{ miles}[/tex]
Total distance traveled by Suzanne in kilometers=[tex]8.25\times1.6\text{ kilometers}=13.2\text{ kilometers}[/tex]
Total distance traveled by Suzanne in kilometers[tex]=13.2\text{ kilometers}[/tex]
Which group of 3 cards would allow Carlos to win with 21 points?
A. 6 of clubs, 7 of spades, 8 of diamonds
B. Ace of hearts, Ace of diamonds, 10 of hearts
C. Jack of clubs, 7 of clubs, 4 of spades
D. 8 of clubs, 9 of hearts, 9 of spades
Answers
☺☺☺☺
Solve the system
4x+2y=7 y=5x
A. (1,5)
B. (3,15)
C. (2,10)
D. (0.5,2.5)
Answers
Replace 'y' with '5x'
4x + 2(5x) = 7
4x + 10x = 7
14x = 7
x = 7/14 = 1/2
Sub into y=5x to find 'y'
y = 5(1/2) = 5/2 = 2.5
Answer:
x = 0.5, y = 2.5
how many factors are in the expression x(1+y)^z
a. 3
b.2
c.0
d.1
Answers
There are two factors: ( x ) and (1+y). So option (b) is correct.
To find the factors in the expression ( x(1+y)^z ), we break it down into its components.
1. ( x ) - This is a factor because it's multiplied by the rest of the expression.
2. (1+y) - This is also a factor because it's part of the expression being raised to the power of \( z \).
3. ( z ) - The exponent does not affect the number of factors, as it's simply the power to which the expression is raised.
So, there are two factors: ( x ) and (1+y).
Detailed calculation:
( x ) is multiplied by the entire expression, so it's a factor.
(1+y) is a factor because it's part of the expression.
( z ) is an exponent and doesn't count as a separate factor.
Thus, the answer is (b) 2.
What is the quotient of d-2 / d4-6d3+d+17=
A. d3− 4d2 + 9d + 35 B. d3− 8d2 + 17d –17 C. d3− 8d2 + 16d − 31 R 79 D. d3– 4d2– 8d –15 R–13
Answers
We have (d⁴-6d³+d+17) ÷ (d-2)
Step 1:
Multiply d by a term that would give result of d⁴
The aim of this working out is to eliminate on one term at a time, starting from the highest polynomial
We have d⁴ ÷ d = d³ (Write this on the top)
Step 2:
Multiply back d³ by (d - 2) to get d⁴ - 2d³
Step 3:
Then subtract (d⁴ - 2d³) from (d⁴ - 6d³ + 0d² + d + 17)
Notice that the term d⁴ is now eliminated
We have (-6d³ - -2d³) = (-4d³)
Then pull down the third term which is 0d²
We have the sum -4d³ + 0d²
Step 4:
Divide (-4d³) by (d) to get (-4d²)
Write (-4d²) next to (d³) on the top
Step 5:
Then multiply back (-4d²) by (d - 2) to get (-4d³+8d²)
Step 6:
Subtract (-4d³+8d²) from (-4d³ + 0d²) to get (-8d²) then pull down the next term (d) to make the sum (-8d² + d)
Notice that the term -4d³ is now eliminated
Step 7:
Divide -8d² by d to get -8d
Write -8d on the top, next to d³ - 4d²
Step 8:
Multiply back (-8d) by (d - 2) to get (-8d² + 16d)
Step 9:
The subtract (-8d² + 16d) from (-8d + d) to get (-15d + 17)
Notice that the term (-8d²) has been eliminated
The last step is to divide -15d by d to get -15
Write -15 on the top next to the sum d³ - 4d² - 8d
Then multiply -15 by (d-2) to get -15d + 30
Subtract (-15d + 30) from (-15d + 17) to get a REMAINDER of 13
The answer is d³ - 4d² - 8d - 15 R 13
OPTION D
Answer:
Step-by-step explanation:
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F(x, y, z) = xe^(5yz) find the gradient of f
Answers
[tex]\nabla F(x,y,z)=\left\langle e^{5yz},5xze^{5yz},5xye^{5yz}\right\rangle[/tex]
Find f. f '''(x) = cos(x), f(0) = 4, f '(0) = 2, f ''(0) = 6
Answers
f'''(x) = Cos(x)
f''(x) = sin(x) + 6 after f''(0) = 6
f'(x) = -cos(x) + 6x + 2 after f'(0) = 2
f(x) = -sin(x) + 3x^2 + 2x + 4 after f(0) = 4
Solve for x.
5x - 1 = 26
x = 27/5
x = 5
x = -5
Answers
5x = 27
x = 27/5
Answer: A) or the first option ✅
Each newborn baby has a probability of approximately 0.49 of being female and 0.51 of being male. for a family with four children, let x = the number of children who are girls.
Answers
I believe this problem has 3 questions:
a. Explain why the three conditions are satisfied for X to have the binomial distribution.
b. Identify n and p for the binomial distribution.
c. Find the probability that the family has two girls and two boys.
Answers:
a. First because there are only 2 possible outcomes for
each birth: male or female. Hence a binomial distribution.
Second, because the probability of giving out a girl is
constant: 0.49 for each birth.
Third, the probability of a giving out a girl does not depend
on whether or not there is already a boy or a girl in the family.
b. The n is the total number of children, so n = 4
While the p is the success of being a girls, so P = 0.49
c. We use the binomial probability equation:
P (X) = nCx * p^x * q^(n-x)
P(X=2) = 4!/(2!2!) * 0.49^2 * 0.51^2 = 0.3747
Final answer:
Inheritance of sex chromosomes in babies is determined by the father, leading to a 50:50 chance of a male or female child. The likelihood of a newborn being female is around 0.49.
Explanation:
Sex Inheritance: The sex of a baby is determined by the father, who has a 50 percent chance of passing on a Y chromosome resulting in a male child. If the baby inherits two X chromosomes, it will be female. This 50:50 chance applies to each baby. For a family with four children, let x = the number of girls, the probability of a child being female is approximately 0.49.
There are currently 3 students signed up for a trip. The van can transport only 7 students.
Which graph shows all the possible values for the extra number of students that need to sign up so that more than one van is needed to transport them?
Number line with closed circle on 4 and shading to the right.
Number line with closed circle on 4 and shading to the left.
Number line with open circle on 4 and shading to the right.
Number line with open circle on 4 and shading to the left.
Answers
Solution:
Call x the number of students that need to sign up so that more than one van is neeed to transport them, then:
x + 3 > 7
=> x > 7 - 3
=> x > 4
That is the numbers greater than 4, without including the number 4.
That, in the number line, is the set of numbers to the right of 4, and the open circle is used to indicate that the number 4 is not included (a closed circle would indicate that the number 4 is included, which is not the case).
Answer:
what he said up so he can get brainliest
Step-by-step explanation:
Lucy is thinking of a number.The number is greater than two hundred twenty- five. her number is less than 2 hundred 2 tens 7 ones. what is lucy's number?
Answers
I am trying to add more characters
on monday your bank account was balanced -$12.58 because you didn't realize this, you wrote a check for -$30.72 for groceries what is the new balance in your checking account?
Answers
u then wrote a check for 30.72....
so ur new balance is :
-12.58 + (- 30.72) =
-12.58 - 30.72 =
- 43.30 <==
The new balance after I bought groceries worth of 30.72 dollars is -43.3 dollars.
What are he types of sign distributions?Sign distributions are of four types they are:- negative distributed to negative results in positive and positive distributed positive results to positive. Negative distributed to positive or vice versa results in negative.
According to the given question on Monday I had -12.58 dollars.
I bought some groceries worth -30.72 dollars which will also be deducted from my bank account.
∴ The final balance I have is ( -12.58 - 30.72) dollars which is,
= -43.3 dollars.
learn more about sign distributions here :
https://brainly.com/question/28608950
#SPJ5
At a pet store, Davina counted 12 parrots out of 20 birds. What is an equivalent ratio of parrots to birds in the pet store?
Answers
How many grams are in 1.40 moles of sodium chloride (NaCl)?
Answers
The molar mass of NaCl is 58.44 g / mol. So we can calculate the mass by simply multiplying the number of moles and molar mass. That is:
mass = 1.40 moles * 58.44 g / mol
mass = 81.82 grams
Final answer:
To find the mass in grams of 1.40 moles of sodium chloride, multiply the number of moles by the molar mass of NaCl, which is 58.44 g/mol, resulting in 81.816 grams.
Explanation:
To calculate the mass of 1.40 moles of sodium chloride (NaCl), we will use the molar mass of NaCl as a conversion factor. The molar mass of NaCl is 58.44 grams per mole. Here's how you can do the conversion:
Find the molar mass of NaCl: 22.99 g/mol (Na) + 35.45 g/mol (Cl) = 58.44 g/mol (NaCl).
Multiply the number of moles of NaCl by the molar mass to get the mass in grams: 1.40 moles × 58.44 g/mol = 81.816 grams of NaCl.
So, 1.40 moles of sodium chloride weigh 81.816 grams.
If an item is 1/3 off, to calculate the final price simply divide the original price by 3.
true or false
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and then multiply the price by 100 to get the total
What is the least possible sum of 24
Answers
A certain recipe calls for 2 3/4 cups of sugar.if the recipe is to be doubled, how much sugar should be used
Answers
Over a 12 hour period from 8 pm to 8 am the tempeture fell a steady rate 8°F to 16°F if the temperature fell at the same rate every hour what was the temperature at 4 am
Answers
12 hour period from 8pm to 8am
temperature dropped from 8°F to 16°F from 8pm to 8am
We need to find temperature at 4 am.
We can start by setting up points:
8pm is are starting point with 8°F, we can express it as (0,8), 0 represents initial time from 0 to 12 hour span.
8am is the ending point with 16°F, we can express it as (12,16), 12 represents the end time of 0 to 12 hours span.
We will use these points to find slope.
slope=m=(16-8)/(12-0)=8/12=2/3
Now, we can set up an expression to find any temperature at a specific time. Aslo, x represents the hours not the the specific time of 4am. We will use 8 since 4am is the 8th hour of the 12 hour span. Using slope of 2/3 and the y intercept of (0,8) since we were already at 8°F at the initial time of 0 we have the function:
f(x)=2/3x+8
f(8)=2/3(8)+8= 40/3≈13.3°
Find equations for the horizontal and vertical tangent lines to the curves
Answers
The shortest side of a right triangle measures 11m. The lengths of the other two sides are consecutive integers. Find the lengths of the other two sides.
Answers
Then (11 m)^2 + x^2 = sum of the squares of the 2 shortest sides
= (x+1)^2
121 + x^2 = x^2 + 2x + 1. Then 121 = 2x + 1, and 2x = 120, or x = 60.
Then the hyp. has length 60+1= 61.
We must check these results. Using the Pyth. Thm. (a^2 + b^2 = c^2),
11^2 + 60^2 = 61^2
121 + 3600 = 3721 This is true, so our answers are correct.
The longer side is 60 and the hyp. has length 61.
In a fraternity with 37 members, 18 take mathematics, 5 take both mathematics and art history, and 8 take neither mathematics nor art history. How many take art history but not mathematics?
Answers
fraternity of 32 members with 8 taking neither math or physics
Of the 24, 5 took both. 24 - 5 = 19 taking one. 18 taking just Math.
Just 1 took only Physics.
checking our answer:
just Math 18
just Physics 1
both 5
neither 8
total 32
just physics = 32 - 8 -5 -18 = 1
Answer:
Fraternity of 32 members with 8 taking neither math or physics
Of the 24, 5 took both. 24 - 5 = 19 taking one. 18 taking just Math.
Just 1 took only Physics.
Step-by-step explanation:
sales price is 67.50 sales tax is 5% how much is the total amount paid
Answers
x/67.5=5/100
cross multiply
100x=337.5
divide 100
x=3.375 or 3.38
67.50+3.38= 70.88
Please help me on 3 and 4 please thanks ❤️
Answers
Since we know that there are 12 inches in a foot, and 12 feet, we have
[tex] \frac{12inches}{1foot} * 12feet=144 inches[/tex]
Feel free to ask with any further questions!