Let
[tex]\displaystyle y(x) = \sum_{n=0}^\infty a_nx^n = a_0 + a_1x + a_2x^2 + \cdots[/tex]
Differentiating twice gives
[tex]\displaystyle y'(x) = \sum_{n=1}^\infty na_nx^{n-1} = \sum_{n=0}^\infty (n+1) a_{n+1} x^n = a_1 + 2a_2x + 3a_3x^2 + \cdots[/tex]
[tex]\displaystyle y''(x) = \sum_{n=2}^\infty n (n-1) a_nx^{n-2} = \sum_{n=0}^\infty (n+2) (n+1) a_{n+2} x^n[/tex]
When x = 0, we observe that y(0) = a₀ and y'(0) = a₁ can act as initial conditions.
Substitute these into the given differential equation:
[tex]\displaystyle \sum_{n=0}^\infty (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^\infty a_nx^n = 0[/tex]
[tex]\displaystyle \sum_{n=0}^\infty \bigg((n+2)(n+1) a_{n+2} - a_n\bigg) x^n = 0[/tex]
Then the coefficients in the power series solution are governed by the recurrence relation,
[tex]\begin{cases}a_0 = y(0) \\ a_1 = y'(0) \\\\ a_{n+2} = \dfrac{a_n}{(n+2)(n+1)} & \text{for }n\ge0\end{cases}[/tex]
Since the n-th coefficient depends on the (n - 2)-th coefficient, we split n into two cases.
• If n is even, then n = 2k for some integer k ≥ 0. Then
[tex]k=0 \implies n=0 \implies a_0 = a_0[/tex]
[tex]k=1 \implies n=2 \implies a_2 = \dfrac{a_0}{2\cdot1}[/tex]
[tex]k=2 \implies n=4 \implies a_4 = \dfrac{a_2}{4\cdot3} = \dfrac{a_0}{4\cdot3\cdot2\cdot1}[/tex]
[tex]k=3 \implies n=6 \implies a_6 = \dfrac{a_4}{6\cdot5} = \dfrac{a_0}{6\cdot5\cdot4\cdot3\cdot2\cdot1}[/tex]
It should be easy enough to see that
[tex]a_{n=2k} = \dfrac{a_0}{(2k)!}[/tex]
• If n is odd, then n = 2k + 1 for some k ≥ 0. Then
[tex]k = 0 \implies n=1 \implies a_1 = a_1[/tex]
[tex]k = 1 \implies n=3 \implies a_3 = \dfrac{a_1}{3\cdot2}[/tex]
[tex]k = 2 \implies n=5 \implies a_5 = \dfrac{a_3}{5\cdot4} = \dfrac{a_1}{5\cdot4\cdot3\cdot2}[/tex]
[tex]k=3 \implies n=7 \implies a_7=\dfrac{a_5}{7\cdot6} = \dfrac{a_1}{7\cdot6\cdot5\cdot4\cdot3\cdot2}[/tex]
so that
[tex]a_{n=2k+1} = \dfrac{a_1}{(2k+1)!}[/tex]
So, the overall series solution is
[tex]\displaystyle y(x) = \sum_{n=0}^\infty a_nx^n = \sum_{k=0}^\infty \left(a_{2k}x^{2k} + a_{2k+1}x^{2k+1}\right)[/tex]
[tex]\boxed{\displaystyle y(x) = a_0 \sum_{k=0}^\infty \frac{x^{2k}}{(2k)!} + a_1 \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}}[/tex]
A father and his young toddler are walking along the sidewalk. For every 3 steps
the father takes, the son takes 5 steps just to keep up. What is the ratio of
the number of steps the father takes to the number of steps the son takes? Add
labels to the columns of the table and place the ratio into the first row of
data. Add equivalent ratios to build a ratio table.
Answer:
Ratio 5:3
Step-by-step explanation:
5:3 equivalent fraction are 10:6, 15:9
Katie invested a total of $6000 part at 3% simple interest and part at 4% simple interest at the end of one year the investment had earned $216 interest how much was invested at each rate?
Answer:
$3600 at 4%$2400 at 3%Step-by-step explanation:
Let x represent the amount invested at 4% (the higher rate). Then the amount invested at 3% is (6000-x), and the total interest earned is ...
0.04x + 0.03(6000 -x) = 216
0.01x +180 = 216 . . . . simpilfy
0.01x = 36 . . . . . . . . subtract 180
x = 3600 . . . . . . . . multiply by 100
6000-x = 2400
Katie invested $3600 at 4% and $2400 at 3%.
A factory made 900 jars of peanut butter. 35% of the jars contained creamy peanut butter. How many jars of creamy peanut butter did the factory make?
Answer:
315 jars
Step-by-step explanation:
This question is merely another way of writing: what is 35% of 900?
Now, when we look at it that way, it's easy to solve.
So, the way to solve is to either multiply 35% by 900, or just interpret that as 0.35 times 900. Anyway, the result is 315.
I hope this helped a bunch! Tell me if you need any further assistance...
( :
solve pls brainliest
Answer:
circle, triangle, rectangle
Answer:
the first cross section would make a circle (option 2, the second would make a triangle (option 6), and the last one would make a rectangle (option 1). hope this helps!
Write the value of the underlined digit in words 5.3479? 7 being the underlined number.
The place value of 7 in words in 5.3479 is seven thousandths
The given number is 5.3479
The underlined digit has been identified as 7
The place values of each of the digits in 5.3479 are:
5 = 5 units
3 = 3 tens
4 = 5 hundredths
7 = 7 thousandths
9 = 9 ten thousandths
Therefore, the place value of 7 in words in 5.3479 is seven thousandths
Learn more here: https://brainly.com/question/25750254
anna is 5ft tall and has a shadow that is 3ft long. she is standing next to the stop sign if the sign’s shadow is 4.5 ft long how tall is the stop sign
Answer:
6.5 feet
Step-by-step explanation:
Anna's shadow is 2 feet shorter than her. So, therefore, the stop sign is 2 feet taller that its shadow.
Derrick's dinner bill, including an 18% tip, is $15.34.
What was the amount of the bill before the tip?
Enter your answer in the box. I WILL GIVE BRANLIEST IF ASWERED CORRECTLY
Answer:
The amount before the tip was $12.58
Step-by-step explanation:
Answer:
The amount before the tip was $12.58
Step-by-step explanation:
WILL GIVE BRAINLIEST ‼️ The graph represents the piecewise function:
Answer:
[tex]f(x) = \left \{ {{x + 3} \quad \text{if} \ -3 \leq x < -1\atop {5} \quad \text{if} \ -1 \leq x \leq1} \right.[/tex]
Step-by-step explanation:
You can find the first function if you know the general form of a line:
[tex]y = mx + b[/tex]
where [tex]m[/tex] is the slope and [tex]b[/tex] the intercept point with the [tex]y[/tex]-axis.
So in the first case the slope is 1 because the next formula
[tex]m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2-0}{-1+3} = 1[/tex]
And you can find the interception point if extend the line, then in our case is 3.
So the function is:
[tex]f(x) = x + 5[/tex]
In the other case is constant function that's mean that the function has the next form
[tex]f(x) = k[/tex]
Where [tex]k[/tex] is a constant.
So you only have to observe what is the value for each point in the interval, in this occasion is 5.
So the final answer is:
[tex]f(x) = \left \{ {{x + 3} \quad \text{if} \ -3 \leq x < -1\atop {5} \quad \text{if} \ -1 \leq x \leq1} \right.[/tex]
Determine the value of x. Round to the tenths place. 30/4 65/x
wow that's really hard... :) greetings from South Korea My name is Naoyoung nei The abbreviation of Nanda is Naoyoung nei And the name DA is the name of my father and mother
What is the unit rate of km per hour in the example below?
You can walk 7/10 km in 1/4 hours.
Maybe 28kms in 1 hour
Or 2,8kms because 7/10 is 0,7km
What is 3/1,000 as a decimal
Answer: 3/1000 as a decimal is 0.003.
Explanation;
3/10 = 0.3
0.3/10 = 0.03
0.03/10 = 0.003
3plss help...................................
How much is f(x) = 3(4)x
Answer:
f ( x ) = 12 x
Step-by-step explanation: 3x4=12 so that makes it f(x)=12x
Dan received 64 dollars for his birthday. He went to a sporting goods store
and bought a baseball glove, baseball, and bat. He had 31 dollars left over,
how much did he spent on the baseball gear?
Answer:
the answer is 33 bucks because you have subtract the 64 -31 =33
Multiply.
3 1/2 times 2 2/3
Answer:
Step-by-step explanation:
Two teams are going on a trip to play a game. There will be
2 buses to take the students. On each bus, there are 60
seats. Teachers and parents will sit in 10 of the seats on
each bus
How many seats are available for students on the buses?
No links
increase 400 in the ratio 3:5
Answer:
250.
Step-by-step explanation:
3 + 5 = 8. => (3/8) * 400 = 3 * 50 = 150. => (5/8) * 400 = 5 * 50 = 250.
Which
graph shows the solution to the
system of linear inequalities?
y > 2/3x + 3
y < - 1/3x + 2
-
Answer:there is no graphs
Step-by-step explanation:
poop
Graph the function below, to estimate the practical domain of the function.
=−0.0150912++6
A. 0≤≤71.5
B. 0≥≥71.5
Two decimal for 2597.58333
Answer:
2597. 58000 and this is the ans
math help me please n
Answer:
Step-by-step explanation:
21x1/12=1.75 cm
1.75x10=17.55 mm
17.55 mm
i need help pwease help me i give brainly
Answer:
the answer is A
Step-by-step explanation:
⎩
⎪
⎪
⎨
⎪
⎪
⎧
y=−2x−5
y=2x−2
Answer:
This doesn't make since.
Step-by-step explanation:
The vector makes angles
[tex]Sin^- {}^{1} (\frac{ \sqrt{2146} }{65} ) \: and \: sin {}^{ - 1} ( \frac{12}{13} )[/tex]
with the X and Y axes.Find the Sin Ratio of the angle that is makes with Z-axis.
Answer:
To know the answer refer to attachment.
Note: There are 3 attachments
Scroll side to see the 3rd attachment
[tex]\Large\textsf{Hope \: It \: Helped}[/tex]
Can I get help please..................
Sara bought 3/4 of a pound of bananas and 3/5 of a pound of pears. How much fruit did Sara buy?
Answer:
27/20
Step-by-step explanation:
¾ + ³/5= 27/20. .......................
need help with these two math problems
Answer:
y = (x - 1)(x - 3)
y = (x + 2)^2
Step-by-step explanation:
3)
Has roots at x = {1, 3}
y = (x - 1)(x - 3)
4)
Has double root at x = -2
y = (x + 2)^2
Explain how each location given in the table is related to the origin.
Refer to the attachment
Matrix fractions 1/2+2/6. I don't understand the steps in this.
Answer: 5/6
Step-by-step explanation:
You have to multiply the denominator and numerator by 3 to get the same denominato. Then you ONLY add the numerator. 3/6 + 2/6 = 5/6.
How many different 3 topping pizzas can be made if 10 different toppings are available?
9514 1404 393
Answer:
120
Step-by-step explanation:
We assume the order of the toppings does not matter. Then the number of interest is the number of combinations of 10 toppings taken 3 at a time. That is given by ...
10C3 = C(10, 3) = 10!/(3!(10 -3)!) = 10·9·8/(3·2·1) = 120
120 different 3-topping pizzas can be made if 10 toppings are available.
_____
Additional comment
If order matters, then 10 choices can be made for the first topping, 9 for the second, and 8 for the third, for a total of 10×9×8 = 720 different pizzas.
Assume that the finishing times in a New York City 10-kilometer road race are normally distributed with a mean of 55 minutes and a standard deviation of 10 minutes. Let X be a randomly selected finishing time.
a. What percent of finish times was higher than 72 minutes? (Give your answers rounded to two decimal places)
b. What percent of finish times was between 52 and 70 minutes? (Give your answers rounded to two decimal places)
c. What is the 40th percentile finish time? (Round your answer to nearest tenth)
d. What is the 95th percentile finish time? (Round your answer to nearest tenth)
Using the normal distribution, it is found that:
a) 4.46% of finish times was higher than 72 minutes.
b) 55.11% of finish times was between 52 and 70 minutes.
c) The 40th percentile of finish times is 52.5 minutes.
d) The 95th percentile of finish times is 71.45 minutes.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.In this problem:
The mean is of 55 minutes, hence [tex]\mu = 55[/tex].The standard deviation is of 10 minutes, hence [tex]\sigma = 10[/tex]Item a:
The proportion is 1 subtracted by the p-value of Z when X = 72, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{72 - 55}{10}[/tex]
[tex]Z = 1.7[/tex]
[tex]Z = 1.7[/tex] has a p-value of 0.9554
1 - 0.9554 = 0.0446
0.0446 x 100% = 4.46%
4.46% of finish times was higher than 72 minutes.
Item b:
Th proportion is the p-value of Z when X = 70 subtracted by the p-value of Z when X = 52, hence:
X = 70:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{70 - 55}{10}[/tex]
[tex]Z = 1.5[/tex]
[tex]Z = 1.5[/tex] has a p-value of 0.9332.
X = 52:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{52 - 55}{10}[/tex]
[tex]Z = -0.3[/tex]
[tex]Z = -0.3[/tex] has a p-value of 0.3821.
0.9332 - 0.3821 = 0.5511
0.5511 x 100% = 55.11%
55.11% of finish times was between 52 and 70 minutes.
Item c:
The 40th percentile is X when Z has a p-value of 0.4, so X when Z = -0.253.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.253 = \frac{X - 55}{10}[/tex]
[tex]X - 55 = -0.253(10)[/tex]
[tex]X = 52.5[/tex]
The 40th percentile of finish times is 52.5 minutes.
Item d:
The 95th percentile is X when Z has a p-value of 0.95, so X when Z = 1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 55}{10}[/tex]
[tex]X - 55 = 1.645(10)[/tex]
[tex]X = 71.45[/tex]
The 95th percentile of finish times is 71.45 minutes.
A similar problem is given at https://brainly.com/question/24663213