Solve for x. Round your answer to the nearest tenth.

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]

Answer:

-3

Step-by-step explanation:

a. Simply put (x+h) wherever you have x.

[tex]f(x+h) = -3(x+h)+7 = -3x +7 -3h[/tex]

b. We have the first part, just subtract

[tex]f(x+h) = -3(x+h)+7 = -3x +7 -3h - (-3x+7) = -3x+7-3h +3x -7 = -3h[/tex]

c. Finally, we divide by h[tex]\frac{-3h}h = -3[/tex]

Answer:

A child psychologist is interested in the number of times a newborn baby’s crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let X = the number of times per week a newborn baby’s crying wakes its mother after midnight. For this example, x = 0, 1, 2, 3, 4, 5.

P(x) = probability that X takes on a value x.

x P(x)0 P(x = 0) = 250

1 P(x = 1) = 1150

2 P(x = 2) = 2350

3 P(x = 3) = 950

4 P(x = 4) = 450

5 P(x = 5) = 150

Step-by-step explanation:

pa follow po pls.

Answer:

LJ< JK<LK

Step-by-step explanation:

for a theorem, the order of the side of the triangle is the same as the order of the opposite angles since for the angles it's K<L<J, you replace each with the opposite side (quick trick, it's the one not containing the letter!)

Answer:

No

No

Yes

Yes

No

Step-by-step explanation:

In this case we are looking at substitution, so any where you find x put three(3) in replace of it.

And remember the answer should be 36,

I find it difficult to type the answer so I will take a picture instead.

Answer:

500 pounds

Step-by-step explanation:

Step-by-step explanation:

To do this divide 2000 by 4.This will give you around 500.

Answer:

0.00706666666

Step-by-step explanation:

Answer:

66

forty-four times 1.5 equals 66

Answer:

You told me to resolve this question

The approximate percentage of the points scored during the second part of the  game is 43%

% score in the second part  = 15/35 * 100

% score in the second part  = 1500/35

% score in the second part = 42.85 ≈ 43%

Hence the approximate percentage of the points scored during the second part of the  game is 43%

Learn more on percentage here: https://brainly.com/question/2236179

Answer:

0.09

Step-by-step explanation:

Answer:

I understand how that feels

Step-by-step explanation:

Answer:

Number 4

Step-by-step explanation:

Side-Angle-Side, it has to be between the two sides.

Answer:

20 three-point scores

61/3=20 R1

Answer:

7

Step-by-step explanation:

61 divided by 8 = 7 remainder 5

you can do 7 3's and 7 5's which would make 7x3 = 21  +  7 x 5 = 35  21 + 35 = 56

61 - 56  = 5 (add 1 to the amount of 5's)

7 3's and 8 5's

Answer:

chu papi

Step-by-step explanation:

The volume of a cuboid toy chest is equal to their product of the length,

width, and height.

Reasons:

The dimensions of the toy chest are;

Toy chest A;

Length = 5 feet

Width = 4 feet

Height = 2 feet

Toy chest B;

Length = 4 feet

Width = 2 feet

Height = 4 feet

Volume of Toy chest A = 5 ft. × 4 ft. × 2 ft. = 40 ft.³

Volume of Toy chest B = 4 ft. × 2 ft. × 4 ft. = 32 ft.³

The volume of the toy chest Mrs. Smith needs = 35 ft.³

The toy chest Mrs. Smith should purchase is Toy chest A, that has a volume of 40 ft.³, which can store items that with a volume of 35 ft.³

The correct option is; Only toy chest A will provide the necessary storage space

Possible question options obtained from a similar question found online are;

Either toy chest have the storage space needed

Neither has the storage space needed

Only toy chest A

Only toy chest B

Learn more about calculating volumes of various shapes here:

https://brainly.com/question/3789826

Answer:

Answer# 1: A. 26

Answer# 2: A. 25

1. What is m + 5n where m = 1 and n = 5?

Substitute 1 for "m" and 5 for "n":

[tex]\sf (1)+5(5)[/tex]

Multiply 5*5 =25

[tex](1)+25[/tex]

Add :

[tex]\sf 26[/tex]

____________________

2. What is 5m + 5n where m = 10 and n = -5?

Substitute 10 for "m" and -5 for "n".

[tex]\sf 5\left(10\right)\:+\:5\left(-5\right)[/tex]

Multiply 5 * 10= 50

[tex]\sf 50+5\left(-5\right)[/tex]

Multiply 5 * -5 = -25

[tex]\sf 50-25[/tex]

Subtract:

[tex]\sf 25[/tex]

___________________________________

Answer:

1386

Step-by-step explanation:

V=πr²*h

V=22/7*7²*9

V=1386

Answer:p=7

Step-by-step explanation:4x7=28

Answer:

four times a number less 10 is equall to 16 ia not

9514 1404 393

Answer:

  A. {2x, x<1; (x+5)/2, x≥1}

Step-by-step explanation:

The left side of the graph (x<1) has positive slope and a y-intercept of 0. Only one answer choice matches: choice A.

Answer:

36

Step-by-step explanation:

8x-20=5x+1

-5       -5

3x=-20+1

+20 +20

3x=21

3  3

x=7

8(7)-20

FED=36

Answer:

49 round beds

Step-by-step explanation:

60 beds in total= 11 not round beds +?

-11 not round beds=--11 not round beds=?

49 round beds=49 round beds

That's the best way I could explain it.

Answer:

[tex]\huge\boxed{z=6;\ x=12.5}[/tex]

Step-by-step explanation:

If two figures are similar, then corresponding sides are in proportion.

Therefore:

[tex]\dfrac{z}{15}=\dfrac{4}{10}\qquad|\text{cross multiply}\\\\10z=(15)(4)\\\\10z=60\qquad|\text{divide both sides by 10}\\\\\boxed{z=6}[/tex]

[tex]\dfrac{x}{5}=\dfrac{10}{4}\qquad|\text{cross multiply}\\\\4x=(5)(10)\\\\4x=50\qquad|\text{divide both sides by 4}\\\\x=\dfrac{50}{4}\\\\\boxed{x=12.5}[/tex]