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]
of the three lines in the graph one has slope m = 1 one has slope m = 2, and one has slope m = 1/5.
Put the lines in order from greatest to least slope.
Answer:
1/5 1 2
Step-by-step explanation:
If 2x + y = 7, what is the value of y when x = 3? A. 1 B. 3 C. 5 D. 7 E. 9
Answer:
the correct answer is A.1
the result is explained below in the photo
Answer:
y = 1
Step-by-step explanation:
2x + y = 7
x = 3
plug in 3 for x
2(3) + y = 7
Multiply 2 * 3
6 + y = 7
Therefor y = 1
By the way this is not my work
Answer:
Use unit converter.
Step-by-step explanation:
geomtry plzz help 15 points
Answer:
False.
Step-by-step explanation:
The sides of a triangle rule asserts that the sum of the lengths of any two sides of a triangle has to be greater than the length of the third side. However, 8+6=14 is less than 15. Hence, the triangle cannot be formed.
Is y=1/2x+7 a function
Rationalize the denominator.
9-10/2
V2+1
Answer:
d. 19√2 -29
Step-by-step explanation:
Multiply numerator and denominator by the conjugate of the denominator.
[tex]\dfrac{9-10\sqrt{2}}{\sqrt{2}+1}=\dfrac{(9-10\sqrt{2})(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}\\\\=\dfrac{9\sqrt{2}-9-10\cdot2+10\sqrt{2}}{2-1}=\boxed{19\sqrt{2}-29}[/tex]
Answer:
Step-by-step explanation:
Which ratio is equivalent to the given ratio? 8: 16 Select each correct answer. 16/31 1/3 1/2 12/24
[tex]8:16\implies \cfrac{8}{16}\implies \cfrac{1}{2}\implies 1:2 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{16}{31}\qquad \bigotimes~\hfill \cfrac{1}{3}\qquad \bigotimes~\hfill \cfrac{1}{2}\qquad \textit{\large \checkmark}~\hfill \cfrac{12}{24}\implies \cfrac{1}{2}\qquad \textit{\large \checkmark}[/tex]
Answer:
1/2. and 12/24
they are both halves just as the ratio is half of the given number
I have more question
Answer:
you would need three wholes because each whole would represent a loaf of pumpkin bread
Sabrina has eight boxes of crayons, with ten crayons in each box. She uses twelve crayons. How many crayons does she
have left?
Answer:
68
Step-by-step explanation:
8 boxes. 10 per one box. 8×10=80
80-12= 68
y>-1/3x+2 x<4 how to solve the system of inequalities by graphing.
Answer:
See belowStep-by-step explanation:
1. Graph each inequality2. Take the intersection of shaded regions as solutionAttached each inequality separately and the solution showing the intersection below.
y > -1/3x + 2
Find the x - and y - intercepts, connect them with dotted line, shade the area above the linex < 4
Draw a dotted line x = 4, shade the area to the left of the lineDrag the tiles to the correct boxes. Not all tiles will be used. What are the domain and the range of function f? range arrowRight domain arrowRight
here are the correct answers :)
Consider the functions f(x) and g(x). The function f(x)=2x-5 and g(x)= -f(x+1)+3. Describe the three transformations that are performed on f(x) to create g(x).
A.
1.
2.
3.
B. Sketch a graph of g(x):
Answer: 3
Step-by-step explanation:
There are three compartments for luggage and each compartment has a capacity of 25 pieces of luggage. The first two compartments are full up and the last one contains 8 pieces of luggage. How many pieces of luggage are there?
Answer:
the first 2 compartments have 25 so 25+25=50, and the last has 8 in it so 50+8=58
What is 1/3÷1/2 math answers?
Answer:
= [tex]\frac{2}{3}[/tex]
Step-by-step explanation:
Result in decimals: 0.66666666666667
x = mutiply
[tex]=\frac{1}{3} x\frac{2}{1}[/tex]
[tex]\frac{1 x 2}{3 x 1 }[/tex]
= [tex]\frac{2}{3}[/tex]
(Hope this helps can I pls have brainlist (crown)☺️)
Express 405 1/3 in simplest radical form.
Answer:
3 cube root of 15
Step-by-step explanation:
theres your simple radical form
8(152/3) is 405 1/3 in simplest radical form.
What is simplification?Simplifying procedures is one way to achieve uniformity in job efforts, expenses, and time. It reduces diversity and variation that is pointless, harmful, or unneeded. Parenthesis, exponents, multiplication, division, addition, and subtraction are all referred to as PEMDAS. The order of the letters in PEMDAS informs you what to calculate first, second, third, and so on, until the computation is finished, given two or more operations in a single statement.
405 1/3 is equivalent to (405*3 + 1)/3 = 1216/3.
To express 1216/3 in simplest radical form, we can first simplify it by dividing both the numerator and denominator by their greatest common factor, which is 8.
1216/3 = (8*152)/3 = 8(152/3)
Now, we can see that 152/3 cannot be simplified any further, so the simplest radical form of 405 1/3 is: 8(152/3)
Learn more about Simplification here:
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In what time will $1000 amount to $1331 at 10% per annum, compounded annually?
3 Years
Step-by-step explanation:Compound interest is a multiplier that affects the new value each time.
If the compound interest is 10% per annum, the multiplier each year is 1.1.
Year 1: $1000 * 1.1 = $1100
Year 2: $1100* 1.1 = $1210
Year 3: $1210* 1.1 = $1331
This means that it will take 3 years for $1000 to amount to $1331 at 10% per annum.
Alternatively, you can use a logarithm. The calculation can be written as:
1000 * [tex]1.1^n[/tex] = 1331, where n is the number of years.
Rearranged, this gives [tex]1.1^n[/tex] = 1331/1000 or [tex]1.1^n[/tex] = 1.331.
A logarithm can work out the value of n, and is written in the form log1.1(1.331), which gives a value of 3.
Consider the expression 8ab + 3b + 16 - 4a. (a) How many terms are there? (b) How many factors are in the first term? Identify them. (c) Which term is a constant? pls do this fast
a. There are 4 terms in the expression
b. There are 3 factors in the first term.
c. The constant is 16.
8ab + 3b + 16 - 4a
A term is any signed number, a variable, or a constant multiplied by a
variable or variables.
Therefore, the terms are 8ab, 3b, 16, and -4a. The terms are 4 in numbers.
The first term is 8ab .The first term has the factors 8 , a and b. This means the first term have 3 factors.
The term that is a constant is 16
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Jax Incorporated reports the following data for its only product. The company had no beginning finished goods inventory and it uses
absorption costing
Sales price
$ 57.50 per unit
Direct materials
$ 10.50 per unit
Direct labor
$ 8.00 per unit
Variable overhead
$ 12.50 per unit
Fixed overhead
$ 1,237,500 per year
1. Compute gross profit assuming (a) 75,000 units are produced and 75,000 units are sold and (b) 110,000 units are produced and
75,000 units are sold.
2. By how much would the company's gross profit increase or decrease from producing 35,000 more units than it sells?
Answer is complete but not entirely correct.
Complete this question by entering your answers in the tabs below.
Required 1
Required 2
Compute gross profit assuming (a) 75,000 units are produced and 75,000 units are sold and (b) 110,000 units are produced
and 75,000 units are sold.
(a) 75,000 Units (b) 110,000 Units
Produced and
Produced and
75,000 Units Sold 75,000 Units Sold
Sales
4,312,500
4,312,500
The gross income or profit of the company is the difference between the
total revenue and the cost of items produced.
The values for the gross profits are;
(a) $750,000(b) $ -335,0002. The profit of the would decrease by $1,085,000Reasons:
The gross profit is given by the equation;
Gross profit = Net revenue - Cost of items sold
The net revenue for 75,000 units = $57.50 × Sales price per unit
∴ The net revenue for 75,000 units = $57.50 × 75,000 = $4,312,500
Cost of production = 75,000 × Per unit cost of Materials plus Labor plus variable overhead + Fixed overhead
Cost of production = 75,000 × (10.50 + 8.00 + 12.50) + 1,237,500 = 3,562,500
The cost of producing 75,00 units per year = $3,562,500
Gross profit = $4,312,500 - $3,562,500 = $750,000
(b) When the number of units produced = 110,000, we have;
Cost of production = 110,000 × (10.50 + 8.00 + 12.50) + 1,237,500 = 4,647,500
Gross profit = $4,312,500 - $4,647,500 = $ -335,000 (a loss of 335,000)
2. By producing 35,000 units more than it sells, we have;
Cost > Revenue, therefore, the gross profit decreases
Let, X, represent the number of units sold, we have;
The number of units produced = X + 35,000
Which gives;
Cost of production = (X + 35,000) × (10.50 + 8.00 + 12.50) + 1,237,500 =
31·X + 2,322,500
Net revenue = X × 57.50 = 57.50·X
The profit = 57.50·X - (31·X + 2,322,500) = 26.5·X - 2,322,500
When X = 75,000, we have;
Gross profit = 26.5 × 75,000 - 2,322,500 = -335,000
Therefore;
The gross profit will decrease by 750,000 - (-335,000) = 1,085,000
The gross profit of the company will would decrease by $1,085,000, if it
sells 75,000 units and produces 110,000, which is 35,000 more units than
is sells.
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What is the meaning of the point shown on this graph?
Mysha’s Height
A: Mysha was 12 inches tall at age 2 1/2 years
B. Mysha was 30 inches tall at age 1 year
C: Mysha was 12 inches tall at the ago 30 months
D: Mysha was 30 inches tall at age 12 years
11/6= 1/p
Pls help!!
Answer:
p = 6/11
Step-by-step explanation:
11/6 = 1/p
=> 11p = 6
=> p = 6/11
Hoped this helped.
what is x over 5 = -3 find x value
5*-3=-15
x=-15
hope this helps
Answer:
x = -15
Step-by-step explanation:
[tex]\frac{x}{5}=-3~(Given)\\\\5(\frac{x}{5})=5(-3)~(Multiply~5~on~both~sides)\\\\x=-15~(Simplify)[/tex]
Solve. 3(x-6) - 8x = -2 + 5(2x+1)
Answer: x = -1.4
Step-by-step explanation:
Answer: x = -7/5
Step-by-step explanation: (3)(x)+(3)(−6)+−8x = −2+(5)(2x)+(5)(1) 3x+−8x+−18 = 10x+−2+5 −5x+−18 = 10x+3 −5x−18-10x = 10x+3-10x −15x−18+18 = 3+18 −15x/-15=21/-15 x = -7/5
HELP WITH GEOMETRY HOMEWORK ASAP!!!
Answer:
26°
Step-by-step explanation:
90°+64°+x°=180° (Sum of all the angles in a right angled triangle=180°)
154°+x°=180°
x°=180-154°
x°=26°
Write a polynomial function with given zeros √8, √10
Answer:
y = x^2 - (sqrt(8) + sqrt(10))x + sqrt(80)
Step-by-step explanation:
we do y = (x- sqrt8)(x -sqrt10) and simplify
What is the measurement of the angle shown below?
Answer:
The correct answer is (c) 30°
Hope it helps
Write the equation of a parabola that has a vertex of (-2,5.5) and passes through the point (-2.5,2.5).
The equation of a parabola that has a vertex of (-2, 5.5) and passes through the point (-2.5, 2.5) is [tex]y=-12(x+2)^2+5.5[/tex]
The vertex form of the equation of a parabola is:
[tex]y=a(x-h)^2+k[/tex]
where (h, k) is the vertex of the parabola
(x, y) is the point that the parabola passes through
The vertex, (h, k) = (-2, 5.5)
The point, (x, y) = (-2.5, 2.5)
Substitute h = -2, k = 5.5, x = -2.5, and y = 2.5 into the equation to solve for a.
[tex]2.5=a(-2.5-(-2))^2+5.5\\\\2.5=a(-2.5+2)^2+5.5\\\\2.5-5.5=a(-2.5+2)^2\\\\-3=a(-0.5)^2\\\\-3=0.25a\\\\a = \frac{-3}{0.25} \\\\a = -12[/tex]
Therefore, the equation of the parabola is:
[tex]y=-12(x-(-2))^2+5.5\\\\y=-12(x+2)^2+5.5[/tex]
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: Một khu đất 20 ha người ta trồng hai loại cây ngô và khoai. Năng suất của ngô tính theo x ha được (5-x/4) tấn trên 1 ha và năng suất của khoai tính theo y ha được (7-y/4) tấn trên 1 ha. Xác định diện tích trồng từng loại cây để sản lượng thu được là lớn nhất.
jelp pls with 2 work pps brainly
Answer:
The underline one is 40.
Missing numbers. 35, 40 45, 50, 55, 60, 65, 70.
Step-by-step explanation:
Add 5 rule
Answer:
5. The tens place.
6. 35, 40, 45, 50, 55, 60, 65.... (And so on)
Step-by-step explanation:
Hoped this helped.
express (25 1/2) ^3 in simplest radical form
Step-by-step explanation:
i think this is the answer if it's wrong I'm sorry
A room is 15 feet long and 12 feet wide. A scale drawing of the room is 10 inches by 8 inches. What is the scale of inches in the drawing to inches in the actual room?
Enter the correct answer in the box.
Answer:
The scale is 1:18 for the entire room, 1 inch in the diagram for 18 inches IRL
Step-by-step explanation:
12 inches = 1 foot
Convert the feet to inches, then divide both sides of the ratio by the # of inches in the diagram
So for length:
10:(15*12)
10:180
1:18
And for width:
8:12(12)
8:144
1:18
Answer:
18 inches
Step-by-step explanation:
The question asks for scale of inches in the drawing and the actual room, so first convert 15 ft and 12 ft into inches.
1 foot= 12in x 1ft= 12 inches
15 feet= 12in x 15ft = 180 inches
12 feet= 12in x 12ft = 144 inches
The room is 180 inches long and 144 inches wide.
To find the scale, set up a proportion to the drawing to the room.
[tex]\frac{10}{8} =\frac{180}{144}[/tex]
You can see that the scale is 18 inches because to get from 180 to 10, you multiply by 18, and the same thing as 8, you would multiply 8 by 18 to get to 144.
