You're looking for a solution in the form
[tex]y(x) = \displaystyle \sum_{n=0}^\infty a_nx^n[/tex]
Differentiating, we get
[tex]y'(x) = \displaystyle \sum_{n=0}^\infty na_nx^{n-1} = \sum_{n=1}^\infty na_nx^{n-1} = \sum_{n=0}^\infty (n+1)a_{n+1}x^n[/tex]
[tex]y''(x) = \displaystyle \sum_{n=0}^\infty (n+1)na_{n+1}x^{n-1} = \sum_{n=1}^\infty (n+1)na_{n+1}x^{n-1} = \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n[/tex]
Substitute these for y' and y'' in the differential equation:
[tex]\displaystyle \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n - \sum_{n=0}^\infty (n+1)a_{n+1}x^n = 0[/tex]
[tex]\displaystyle \sum_{n=0}^\infty \bigg((n+2)(n+1)a_{n+2}-(n+1)a_{n+1}\bigg)x^n = 0[/tex]
Then the coefficients of y are given by the recurrence
[tex]\begin{cases}a_0=y(0)\\a_1=y'(0)\\a_{n+2}=\frac{a_{n+1}}{n+2}&\text{for }n\ge0\end{cases}[/tex]
or
[tex]a_n = \dfrac{a_{n-1}}n[/tex]
But we cannot assume that [tex]a_0[/tex] and [tex]a_1[/tex] depend on each other; we can only guarantee that the recurrence holds for n ≥ 1, so that
[tex]a_2=\dfrac{a_1}2 \\\\ a_3=\dfrac{a_2}3=\dfrac{a_1}{3\times2} \\\\ a_4=\dfrac{a_3}4=\dfrac{a_1}{4\times3\times2} \\\\ \vdots \\\\ a_n=\dfrac{a_1}{n!}[/tex]
So in the power series solution, we split off the constant term and we're left with
[tex]y(x) = a_0 + a_1 \displaystyle \sum_{n=1}^\infty \frac{x^n}{n!}[/tex]
so that the fundamental solutions are
[tex]y_1=1[/tex]
and
[tex]y_2=x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\cdots[/tex]
what is 24 subtracted from 8
Hi!
8 - 24 = -(24 - 8) = -16
Answer:
-16
Step-by-step explanation:
8-24=-16
b) What is the 4 times of the sum of 3and9?
Answer:
108
Step-by-step explanation:
sum is a fancy word for add so 3+9=27 and 27*4=108
if the volume of a cube is 2197cm3, find the height of the cube
two interior angles of a trapezium sum up to 250 degrees If the remaining angles are equal find the value
9514 1404 393
Answer:
each is 55°
Step-by-step explanation:
The sum of angles in a trapezium is 360°, so the sum of the remaining two angles is 360° -250° = 110°. Each of those two equal angles will be ...
110°/2 = 55°
If u= 70% and o=5%, what % of scores fall within 3 standard deviations from the mean?
Answer:
"85%" is the right answer.
Step-by-step explanation:
Given:
[tex]\mu = 70[/tex] (%)
[tex]\sigma = 5[/tex] (%)
As we know,
The 99.7% observation fall within the 3rd standard deviation, then
⇒ [tex](\mu \pm \sigma ) = (70-(3\times 5)) \ to \ (70+(3\times 5))[/tex]
[tex]=(70-15) \ to \ (70+15)[/tex]
[tex]=55 \ to \ 85[/tex] (%)
Thus the above is the correct solution.
Consider the piecewise function shown on the graph, which is composed of three different function types. ----- IMAGE ATTACHED BELOW
Match each piece of the function with its domain.
Answer:
[tex]Domain = (-\infty,-2)[/tex] --- quadratic function
[tex]Domain = (-2,6)[/tex] --- linear function
[tex]Domain = (6,\infty)[/tex] --- square root function
Step-by-step explanation:
Given
The attached graph
Required
The domain of each function
Starting from the left; the first function is the quadratic function.
The curve of the quadratic function stops at -2 So, the domain is:
[tex]Domain = (-\infty,-2)[/tex]
The straight line that starts at -2 and ends at 6 is a linear function.
So, the domain is:
[tex]Domain = (-2,6)[/tex]
Lastly, the square root function begins at 6. So, the domain is:
[tex]Domain = (6,\infty)[/tex]
Answer:
quadratic = (-∞,-2)
square root = (6,∞)
linear = (-2,6)
Step-by-step explanation:
The starting salaries of individuals with an MBA degree are normally distributed with a mean of $40,000 and a standard deviation of $5,000. What percentage of MBA's will have starting salaries of $34,000 to $46,000
Answer:
The correct answer is "76.98%".
Step-by-step explanation:
According to the question,
⇒ [tex]P(34000<x<46000) = P[\frac{34000-40000}{5000} <\frac{x- \mu}{\sigma} <\frac{46000-40000}{5000} ][/tex]
[tex]=P(-1.2<z<1.2)[/tex]
[tex]=P(z<1.2)-P(z<-1.2)[/tex]
[tex]=0.8849-0.1151[/tex]
[tex]=0.7698[/tex]
or,
[tex]=76.98[/tex]%
I need help with this
To win the game, Elena has to roll an even number first and a number less than 3 second. Her probability of winning is StartFraction 6 over 36 EndFraction.
Answer:
Sum of 6
Sum of 2 or 9
Sum (> 2 but < 5)
Step-by-step explanation:
We are given that :
Elena's probability of winning is 6 /36 = 1/6
And also that Martha's probability if winning is lower Than that of Elena ; Hence, Martha's outcome should be outcjnes whose probability is less than 1/6 (Elena's probability of winning)
Using a sample space that gives the sum of 2 dices.
Recall :
Probability = required outcome / Total possible outcomes
Total possible outcomes for a two dice throw = 6² = 36
Using the sample space attached, we can count the sums from the sample space :
To obtain a sum of 7 :
P(sum 7) = 6 /36 = 1/6
To obtain a sum of 6 :
P(sum 6) = 5 /36
Sum of 2 or 9:
P(sum of 2 or 9). = 5 / 36
Sum > 9 :
P(sum > 9). = 6/36
P Sum (> 2 but < 5) = 5 /36
Correct choices are probability values less than 6/36 which are :
Sum of 6
Sum of 2 or 9
Sum (> 2 but < 5)
Answer:
rolling a sum of 6
rolling a sum of 2 or a sum of 9
rolling a sum that is greater than 2 but less than 5
Step-by-step explanation:
An absolute value function has
A. Curved lines that only increases and decreases.
B. Straight lines that do both increase ,decrease, or stay constant on the same graph
C.Straight line that do both increase and decrease on the same graph
D. Straight lines that only increase or decrease
E. Curved lines that do both increase and decrease on the same graph
Red maple trees can reach heights up to 80 feet. What is the height of the maple tree shown below?
Answer:
h = 54.5
Step-by-step explanation:
Since this is a right triangle, we can use trig functions
sin theta = opp / hyp
sin 35 = h/95
95 sin 35 = h
h=54.48976
Rounding to the nearest tenth
h = 54.5
Answer:
Step-by-step explanation:
[tex]sin 35/h = sin 90/95\\n=54.48976145\\54.5[/tex]
lisa used 880g of a container of sugar to bake a cake and 1/10 of the creaming sugar to make cookies. She then had 3/7 of the container of sugar left. How much sugar was in the container at first
Answer:
At the beginning, there were 2,678.26 grams of sugar in the container.
Step-by-step explanation:
Since Lisa used 880g of a container of sugar to bake a cake and 1/10 of the creaming sugar to make cookies, and she then had 3/7 of the container of sugar left, to determine how much sugar was in the container at first, the following calculation must be performed:
880 + 1 / 10X = 3 / 7X
880 + 0.1X = 0.4285X
880 = 0.4285X - 0.1X
880 = 0.3285X
880 / 0.3285 = X
2,678.26 = X
Therefore, at the beginning there were 2,678.26 grams of sugar in the container.
Serkan teacher regularly buys 75 TL of gasoline in his car every week.
At the end of the 13th week, how much is the total gasoline expenditure made by the serkan teacher?
A)390 B)420 C)900 D)975
Answer:
d
Step-by-step explanation:
75 per week,
after 13 weeks, 75*13 = 975
Muhammad lives twice as far from the school as Hita. Together, the live a total of 12 km
from the school. How far away drom the school does each of them live?
Answer:
Muhammad lives 8 km away from the school.
Hita lives 4 km away from the school.
Step-by-step explanation:
First of all, find a number that, when you double that number and add both numbers, you will get 12. That number is 4. So double 4 and get 8. Then add both to get 12.
Find the area of the sector formed by central angle
θ
in a circle of radius
r
if
θ
=
2
;
r
=
6
m
Answer: 0.2pi
Step-by-step explanation:
1. Find the area of the entire circle
2. Set up a proportion that compares the relationship of the Area of sector and the Area of circle to the Arc measure and the circle measure
3. Solve!
The product of 86 and the depth of the river
Answer:
Step-by-step explanation:
Are you trying to find a variable expression? the product of 86 means multiplication so 86*n or 86n. Other than that I dont understand the question.
Samir estimates the value of Three-fifths times 16.1. Which estimate is reasonable?
3
9
12
15
Answer: 9
Step-by-step explanation:
[tex] \frac{3}{5} \times 16.1 = 9.66[/tex]
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that
Rn(x) → 0.] Find the associated radius of convergence R.
f(x) = 8(1 − x)^−2
show step by step including finding the derivatives.
Recall that for |x| < 1, we have
[tex]\displaystyle \frac1{1-x} = \sum_{n=0}^\infty x^n[/tex]
Differentiating both sides gives
[tex]\displaystyle \frac1{(1-x)^2} = \sum_{n=0}^\infty nx^{n-1} = \sum_{n=0}^\infty (n+1)x^n[/tex]
and multiplying both sides by 8 gives the series for f(x) :
[tex]f(x)=\displaystyle \frac8{(1-x)^2} = \boxed{8\sum_{n=0}^\infty (n+1)x^n}[/tex]
and this converges over the same interval, |x| < 1, so that the radius of convergence is 1.
Each minute Garret is able to run 124 meters. If he has already run 328 meters, what will his total distance be after 11 minutes?
A. 1,692 meters
B. 2,244 meters
C. 3,674 meters
D. 4,972 meters
Answer:
A.
Step-by-step explanation:
124 * 11 = 1364
1364 + 328 = 1,692
Can someten help me find the answer?
Answer:
D
Step-by-step explanation:
It subtracts.
Please help …………………….
9514 1404 393
Answer:
(-3, 3)
Step-by-step explanation:
The blanks are trying to lead you through the process of finding the point of interest.
__
The horizontal distance from T to S is 9 . (or -9, if you prefer)
The ratio you're trying to divide the line into is the ratio that goes in this blank:
Multiply the horizontal distance by 2/3 . (9×2/3 = 6)
Move 6 units left from point T.
The vertical distance from T to S is 6 .
Multiply the vertical distance by 2/3 . (6×2/3 = 4)
Move 4 units up from point T.
__
Point T is (3, -1) so 6 left and 4 up is (3, -1) +(-6, 4) = (3-6, -1+4) = (-3, 3). The point that is 2/3 of the way from T to S is (-3, 3).
Find the probability of 3 success for the binomial experiment with 7 trial and the success probability of 0.3. Then find the mean and standard deviation. Write the formula substitute
the values.
Answer:
[tex]P(x=3)=0.2269[/tex]
Mean=2.1
Standard deviation=1.21
Step-by-step explanation:
We are given that
n=7
Probability of success, p=0.3
q=1-p=1-0.3=0.7
We have to find the probability of 3 success for the binomial experiment and find the mean and standard deviation.
Binomial distribution formula
[tex]P(X=x)=nC_xp^{x}q^{n-x}[/tex]
Using the formula
[tex]P(x=3)=7C_3(0.3)^3(0.7)^{7-3}[/tex]
[tex]P(x=3)=7C_3(0.3)^3(0.7)^{4}[/tex]
[tex]P(x=3)=\frac{7!}{3!4!}(0.3)^3(0.7)^{4}[/tex]
[tex]P(x=3)=\frac{7\times 6\times 5\times 4!}{3\times 2\times 1\times 4!}(0.3)^{3}(0.7)^{4}[/tex]
Using the formula
[tex]nC_r=\frac{n!}{r!(n-r)!}[/tex]
[tex]P(x=3)=0.2269[/tex]
Now,
Mean, [tex]\mu=np=7\times 0.3=2.1[/tex]
Standard deviation, [tex]\sigma=\sqrt{np(1-p)}[/tex]
Standard deviation, [tex]\sigma=\sqrt{7\times 0.3\times 0.7}[/tex]
Standard deviation, [tex]\sigma=1.21[/tex]
Write each set in the indicated form.
If you need to use "
…" to indicate a pattern, make sure to list at least the first four elements of the set.
Answer:
a. Set-builder form: {y | y is a natural number and 12 ≤ y ≤ 15}
Or
{y | y is a natural number and 11 < y < 16}
b. Rooster form: {3, 4, 5 ,6, ...}
Step-by-step explanation:
a. Rooster form: {12, 13, 14, 15}
All four numbers are natural numbers, therefore we would write this set of numbers in set builder form such that they will all have the same property. Thus:
Set-builder form: {y | y is a natural number and 12 ≤ y ≤ 15}
Or as
{y | y is a natural number and 11 < y < 16}
b. Set-builder form: {y | y is a natural number and y > 2}
Since natural numbers are positive integers, this tells us that all values of the set are not less than or equal to 2. Therefore, they are integers that range from 3 and above.
Thus:
Rooster form: {3, 4, 5 ,6, ...}
Which theorem accurately completes Reason A?
hich theorem accurately completes Reason A
Please help!!! Find the domain of the function y = 2 cot(5∕8x).
A) All real numbers except odd integer multiples of 8π∕5
B) All real numbers except 0 and integer multiples of 8π∕5
C) All real numbers except 0 and integer multiples of 4π∕5
D) All real numbers except odd integer multiples of 4π∕5
Answer:
Function 
Step-by-step explanation:
) Domain is defined as the set of possible values of x where function is defined.
For domain,
So,
The value of x is define as
The domain of the function is all real numbers except
The range is defined as all the y values for every x.
So, The range of the function is all real numbers.
2) The general form of the cot function is
Where, Period is
On comparing, B=3
So, The period of the given function is
3) Vertical asymptote is defined as the line which approaches to infinity but never touches the line.
The vertical asymptote is at where function is not defined.
The two vertical asymptote is
Put n=0,
Put n=1,
So, The two vertical asymptote are
Answer:
B.)
Step-by-step explanation:
took test
Students were sampled in order to determine their support for the legalization of gambling in their community. A sample of 150 students were asked whether or not they supported legalization of gambling, and the following results were obtained.
Do You Support? Number Of
YES 40
NO 60
NO OPINION 50
a. The value of the chi-square test statistic equals _____?
b. The number of degrees of freedom associated with this scenario is _____?
Answer:The correct answer is They wanted to impede the sale of alcohol.
Step-by-step explanation:
They had beliefs that alcohol was against Christianity and that it ruins families and since it ruins families it should be prohibited. They eventually managed to win enough support and ban all alcohol which lasted for a few years before the prohibition ended.
HELP!!!!
Which of the following is the absolute value of 6 − 3i?
A) 3i√3
B) 3√5
C) 3√5i
D) 3√3
Answer:
B
Step-by-step explanation:
We want to find the value of:
[tex]\displaystyle |6-3i|[/tex]
Recall that given a complex number z in the form:
[tex]z=a+bi[/tex]
The absolute value of z will be given by:
[tex]\displaystyle |z| = \sqrt{a^2+b^2}[/tex]
We have the complex number:
[tex]6-3i[/tex]
Thus, a = 6 and b = -3.
Then its absolute value will be:
[tex]|6-3i|=\sqrt{(6)^2+(-3)^2}[/tex]
Evaluate:
[tex]\displaystyle |6-3i|= \sqrt{36+9}=\sqrt{45}=3\sqrt{5}[/tex]
Hence, our answer is B.
The following data points represent the number of remote controllers each student in Tria's video game club owns.
Sort the data from least to greatest.
0
0
7
7
4
4
2
2
0
0
1
1
8
8
0
0
10
2
2
5
5
Find the interquartile range (IQR) of the data set.
Use the figure to find x.
Answer:
Step-by-step explanation:
The sides of a 30-60-90 triangle are in the ratio 1:√3:2
The side opposite the 30° angle is (12√6)÷2 = 6/√6.
The side opposite the 60° angle is √3×6/√6 = 6/√2 =3√2.
The sides of a 45-45-90 triangle are in the ratio 1:1:√2
The hypotenuse is 3√2, so the side opposite the 45° angle is 3.
x = 3
solve above question
