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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]

What are you asking?

10(6) + 25 = 60 + 25 = 85

10(2.75) + 25 = 27.5 + 25 = 52.5

10(1.482) + 25 = 14.82 + 25 = 39.82

93 = 10x + 25 || 68 = 10x || x = 6.8

42.1 = 10x + 25 || 17.1 = 10x || x = 1.71

116.25 = 10x + 25 || 91.25 = 10x || x = 9.125

The value of each expression is given below,

B(6)=85

B(2.75)=52.5

B(1.482)=39.82

Value of x for each expression,

B(x)=93; x=6.8

B(x)=42.1; x=1.7

B(x)=116.25; x=9.125

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given expression is B(x)=10x+25.

The value of each expression will b calculated as,

10(6) + 25 = 60 + 25 = 85

10(2.75) + 25 = 27.5 + 25 = 52.5

10(1.482) + 25 = 14.82 + 25 = 39.82

93 = 10x + 25 || 68 = 10x || x = 6.8

42.1 = 10x + 25 || 17.1 = 10x || x = 1.71

116.25 = 10x + 25 || 91.25 = 10x || x = 9.125

To know more about an expression follow

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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:

  B)  102 units

Step-by-step explanation:

The diagonals of a parallelogram bisect each other. This means ...

  BE = ED

  3 -9x = -8x +14

  -11 = x . . . . . . . . . . . . add 9x -14

  BE = 3 -9x = 3 -9(-11) = 3 +99

  BE = 102 units

Answer:

A' = (8,-6)

B' = (9,11)

C' = (-3,-7)

Step-by-step explanation:

When reflecting a point over the y axis, multiply the point's x coordinate by -1. Do this for each point and you have your answer.

Answer:

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

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]

Step-by-step explanation:

Answer:

Enlargement (aka: Dilation)

Step-by-step explanation:

As the triangle's scale factor is 3, The triangle P enlarges by 3 onto triangle Q.

Answer:

dilation by a scale factor by 3 about (1, 0)

Step-by-step explanation:

it is dilation by a scale factor of 3

see the below picture to find the center of dilation formula

(2, 1)  (4, 3) points of transformation

center = (3*2 - 4 / 3 - 1,  3*1 - 3 / 3 - 1)

= ( 6 - 4 / 2,  0)

= (1, 0)

so single transformation is  dilation by a scale factor by 3 about (1, 0)

Answer:p=7

Step-by-step explanation:4x7=28

Answer: The graph I have attached an image of does not represent a function.

Step-by-step explanation:

This is because if we take the vertical line test, which is to draw a vertical line through any point of the x-axis. If the line passes through two points when it is vertical, the line doesn't represnt

a function.

Answer:

The correct answer goes from the 1st quadrant to the 4th quadrant.

Step-by-step explanation:

~Hope this helps!

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:

4 better inbox

Step-by-step explanation:

Answer:

Harper would take the Total Bill after tax and mutiply it by 0.16.

The formula for this would be

Total Bill = x

Tip = 16% -> 16/100 = 0.16

So

0.16x

Answer:

m∠R=76°

Step-by-step explanation:

∠STR=∠RST=(2x+2)° (base angles of isos triangle)

2(2x+2)+(3x+1)=180 (angle sum of triangle)

      4x+4+3x+1=180

               7x+5=180

                    7x=180-5

                        =175

                      x=175÷7

                        =25

Hence, m∠R = (3x+1)°

                     = [3(25)+1]°

                     = (75+1)°

                     = 76°

Answer:

m∠R=76º

Step-by-step explanation:

All angles in a triangle add up to 180º.

The triangle given is an isosceles triangle.

(2x+2)+(3x+1)+(2x+2)=180

Remove parentheses.

2x+2+3x+1+2x+2=180

Collect like terms.

7x+2+1+2=180

Add.

7x+5=180

Subtract 5 from both sides of the equation.

7x=175

Divide 7 from both sides of the equation.

x=25

m∠R=(3x+1)º

3x25+1=76º

m∠R=76º

Hope this helps :)

Step-by-step explanation:

the numbers : x, y

x + y = 65

x = y + 5

we use the second equation in the first equation :

y + 5 + y = 65

2y + 5 = 65

2y = 60

y = 30

x = y + 5 = 30 + 5 = 35

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:

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:

A third is written as ¹/₃

Step-by-step explanation:

The way to understand this is the number on the bottom is the number of pieces being divided into;

So ¹/₃ is 1 whole divided into 3 parts;

So, you need 3 ¹/₃ parts to make a whole

[tex]a \ * \ \frac{b}{c} = \frac{ab}{c} \\\\ 3 \ * \ \frac{1}{3} = \frac{3 \ * \ 1}{3} \\\\ = \frac{3}{3} \\\\ = 1[/tex]

Step-by-step explanation:

the lengths of any line (side, height,...) in A is twice as long as in B.

now, the area of a triangle is

baseline × height / 2

so, two lines are multiplied. that means the factor 2 gets multiplied in 2 times. and that means the area increases by the factor 2×2 = 4

therefore, the area of A is

40.5 × 4 = 162 cm²

Answer:

0.09

Step-by-step explanation:

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:

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:

12/2 = 6

Step-by-step explanation:

2 ways to do it, graph it, or do the slop finding formula y2-y1/x2-x1

answer:

Step-by-step explanation:

I hope she is correct answer

Answer:

Please double check the question

y = 2√2 is not on the unit circle

The exact measure of the angle is 45°. If the terminal side of angle θ passes through a point on the unit circle in the first quadrant where y=2√2,

A Unit circle is defined as a circle having a radius of 1 unit and having its center at the origin.

We know that the terminal side passes through a point of the form (√2/2, y).

Notice that the point is on the unit circle, so its module must be equal to 1, so we can write:

[tex]1=\sqrt{(\dfrac{\sqrt{2}}{2})^2+y^2}[/tex]

[tex]1^2=\dfrac{2}{4}+y^2[/tex]

[tex]1-\dfrac{1}{2}=y^2[/tex]

[tex]Y=\dfrac{1}{\sqrt{2}}[/tex]

We know that y is positive because the point is on the first quadrant.

Now, we know that our point is:

(√2/2, 1/√2)

And we can rewrite:

√2/2 = 1/√2

So the point is:

( 1/√2,  1/√2)

Finally, remember that a point (x, y), the angle that represents it is given by:

θ = Atan(y/x).

Then in this case, we have:

θ = Atan(1/√2/1/√2) = Atan(1) = 45°

If you want to learn more about angles, you can read:

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Answer:

You told me to resolve this question

Answer:

0.00706666666

Step-by-step explanation:

Answer:

yes because that are true