Solve the following ODE by using power serie: (1+x?)y" + 2xy' – 2y = 2. =
Answers
To solve the given ODE using power series, we assume a power series solution and find a recurrence relation for the coefficients. By setting up the recurrence relation and solving it, we can determine the coefficients of the power series solution, which will give us the solution to the ODE.
To solve the given ordinary differential equation (ODE) using power series, we can assume a power series solution of the form:
y(x) = ∑[n=0 to ∞] a_n * x^n
where a_n are the coefficients to be determined.
Differentiating y(x), we get:
y'(x) = ∑[n=0 to ∞] a_n * n * x^(n-1)
= ∑[n=1 to ∞] a_n * n * x^(n-1)
Similarly, differentiating y'(x), we get:
y''(x) = ∑[n=1 to ∞] a_n * n * (n-1) * x^(n-2)
= ∑[n=0 to ∞] a_(n+2) * (n+2) * (n+1) * x^n
Substituting these expressions into the given ODE:
(1 + x^2) * ∑[n=0 to ∞] a_(n+2) * (n+2) * (n+1) * x^n + 2x * ∑[n=1 to ∞] a_n * n * x^(n-1) - 2 * ∑[n=0 to ∞] a_n * x^n = 2
Expanding and rearranging the terms:
∑[n=0 to ∞] a_(n+2) * (n+2) * (n+1) * x^n + ∑[n=0 to ∞] a_(n+2) * (n+2) * (n+1) * x^(n+2) + 2 * ∑[n=1 to ∞] a_n * n * x^n - 2 * ∑[n=0 to ∞] a_n * x^n = 2
Grouping terms with the same power of x:
(a_2 * 2 * 1 + 2 * a_0 - 2 * a_0) * x^0 + (a_3 * 3 * 2 - 2 * a_1) * x + ∑[n=2 to ∞] [a_(n+2) * (n+2) * (n+1) + 2 * a_n * n - 2 * a_(n-2)] * x^n = 2
To satisfy this equation for all values of x, each term coefficient must be equal to zero. So, we can set up the following recurrence relation:
a_2 = 0
a_3 = 0
a_(n+2) * (n+2) * (n+1) + 2 * a_n * n - 2 * a_(n-2) = 0 for n ≥ 2
Solving this recurrence relation will give us the values of a_n for n ≥ 2, which will determine the power series solution y(x).
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Related Questions
a rectangular field is to be covered on three sides leaving a side of length 20m uncovered. if the area of the field is 600m2. how many metres of fencing is required?
Answers
Answer:
580m^2
Step-by-step explanation:
600m^2 -20m=580m^2
Help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
to tiny submit closer picture
Which statement about convergent infinite
geometric series is true?
A- The graph of a convergent infinite geometric
series moves towards infinity.
B- Convergent infinite geometric series sum to a
single value.
C-The graph of a convergent infinite geometric
series curves away from its sum.
D- A convergent infinite geometric series could
have a common ratio of 4.
Answers
Answer:
B) Convergent infinite geometric series sum to a single value
Step-by-step explanation:
An infinite geometric series of the form \(\[ \sum_{r=1}^{\infty} ar^{n-1}\) converges if \(|r|<1\), so the summation of the series will result in a single value if that is true.
please help me answer this question asap
Answers
Answer:
It's quite easy
Step-by-step explanation:
people less than 30 years = frequency of people 0 to 15 + 15 to 30 = 8+15 =23
Therefore there are 23 people less than 30 years old.
pls mark me as brainliest pls.
if f is continuous and ∫ 4 0 f(x)dx = 10, find ∫ 2 0 f(2x)dx.
Answers
Since f is continuous, we can use the substitution u = 2x to get ∫ 2 0 f(2x)dx = (1/2)∫ 4 0 f(u)du.
Therefore, ∫ 2 0 f(2x)dx = (1/2)∫ 4 0 f(u)du = (1/2)(10) = 5.
So the value of ∫ 2 0 f(2x)dx is 5.
Given that f is continuous and ∫₀⁴ f(x)dx = 10, we need to find ∫₀² f(2x)dx. To do this, let's use substitution: let u = 2x. Then, du/dx = 2, and dx = du/2.
Now, we substitute the limits of integration:
When x = 0, u = 2(0) = 0.
When x = 2, u = 2(2) = 4.
The integral becomes ∫₀⁴ f(u) (du/2). To solve, we factor out the constant 1/2:
(1/2) ∫₀⁴ f(u)du.
Since ∫₀⁴ f(x)dx = 10, we can replace the integral:
(1/2) * 10 = 5.
Thus, ∫₀² f(2x)dx = 5.
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The absolute value of (2−7)=
Answers
That because anything in the absolute value that is a negative answer it will always be positive. Like 2-7 which supposed to equal to -5 but when it come in a absolute value the answer will be positive 5.
The absolute value is:
5Work/explanation:
First, we will evaluate 2-7.
It evaluates to -5.
Now, let's find the absolute value of -5 by using these rules:
\(\sf{\mid a\mid=a}\)
\(\sf{\mid-a \mid=a}\)
Similarly, the absolute value of -5 is:
\(\sf{\mid-5\mid=5}\)
Hence, 5 is the answer.Please help I missed school for a while I don't know how to do this math I'm so behind right now
Using the trig functions of sin cos and tan set up the problem the trig function should be equal to a ratio reduce fractions if possible
Answers
The values of the trigonometric identities are;
1. tan X = 10/17
2. tan X = 29/25
3. sin A = 24/25
4. tan C = 15/17
5. sin A = 8/17
6. cos A = 9/41
7. cos Z = 3/4
How to determine the trigonometric fractionsNote that fractions are the part of a whole number.
Also,
sin θ = opposite/hypotenuse
cosθ = adjacent /hypotenuse
tan θ = opposite/adjacent
1. tan X = opposite/ adjacent
tan X = 30/21 = 10/7
2. tan X = 29/25
3. sin A = opposite/hypotenuse
sin A = 24/25
4. tan C = opposite/adjacent
tan C = 30/34
tan C = 15/17
5. sin A = 16/34
sin A = 8/17
6. cos A = adjacent/hypotenuse
cos A = 9/41
7. sin A = 21/35
8. cos Z = 21/28
cos Z = 3/4
Hence, the fractions takes the function of the trigonometric identities.
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A town has a population of 15000 and grows at 4. 5% every year. To the nearest tenth of a year, how long will it be until the population will reach 20700?.
Answers
At a 4.5% annual growth rate, it will take 8.4 years to double the population from 15,000 to 20700.
What is percent?In essence, percentages are fractions with a 100 as the denominator. We place the percent symbol (%) next to the number to indicate that the number is a percentage. For instance, you would have received a 75% grade if you answered 75 out of 100 questions correctly on a test (75/100). A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.
Here,
The initial population=15000
Growth rate=4.5%
Final population=20700
Population growth=15000*4.5%
=675
Let n be the number of years.
Final population=initial population + Population growth*number of years
20700=(15000+675n)
675n=20700-15000
675n=5700
n=5700/675
n=8.4 years
It will take 8.4 years to grow the population from 15000 to 20700 at the growth rate of 4.5% per year.
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Destiny has a bag with 3 green, 4 red, and 5 blue marbles. Destiny will pick 2 marbles from the bag, one at a time without looking. She will not replace a marble before picking the next marble. What is the probability of Destiny picking a red marble, followed by a green marble?
A. 1/12
B. 1/11
C. 7/12
D. 1/2
Answers
Let’s find how many green and red balls are in the bag. 4+3=7
So that means the probability of picking a red marble followed by a green marble is 7/12 (C)
Write the equation of this line in slope-intercept form.
Answers
Answer:
Y = 1/2x + 4
Step-by-step explanation:
The +4 is from the 4 that is in the 0 spot.
The 1/2 is how high the slope goes up.
Therefore, the answer is Y = 1/2x + 4.
A police officer keeps record of 75 drivers at a stop sign.
They observe 21 people who came to a complete stop, 46
who made a rolling stop, and 8 who did not appear to
brake at all. What is the probability that a randomly
selected driver will come to a complete stop? Write your
answer as a percent?
Answers
Answer:
Step-by-step explanation
4 and 5)16.97 or 12√2
6)15.59 or 9√3
7)y = 15.59 or 9√3 x =27
8)17.80
A drawer contains 8 blue socks, 8 black socks, and 4 white socks. Socks are picked at random. Explain why the events picking a blue sock and then another blue sock are dependent. Then find the probability.
Answers
Answer:
because he just wants his socks to wear
Step-by-step explanation:
i am confused on how to solve this because i tried using
system of equations and it was incorrect.
For the piecewise function, find the values h(-10), h(1), h(4) , and h(8) . h(x)=\{\begin{array}{ll} -2 x-16, & { for } x
Answers
The values of the piecewise function h(x) at -10, 1, 4, and 8 are:
h(-10) = 4
h(1) = 5
h(4) = 24
h(8) = 80
To find the values of the piecewise function h(x) at specific points, let's evaluate h(-10), h(1), h(4), and h(8) using the given function definition.
We have the following piecewise function:
h(x) =
-2x - 16 for x < -3
5 for -3 ≤ x < 2
x^2 + 2x for x ≥ 2
Evaluating h(-10):
Since -10 is less than -3, we use the first function definition:
h(-10) = -2(-10) - 16
= 20 - 16
= 4
Evaluating h(1):
Since 1 is greater than or equal to -3 but less than 2, we use the second function definition:
h(1) = 5
Evaluating h(4):
Since 4 is greater than or equal to 2, we use the third function definition:
h(4) = 4^2 + 2(4)
= 16 + 8
= 24
Evaluating h(8):
Since 8 is greater than or equal to 2, we use the third function definition:
h(8) = 8^2 + 2(8)
= 64 + 16
= 80
Therefore, the values of the piecewise function h(x) at -10, 1, 4, and 8 are:
h(-10) = 4
h(1) = 5
h(4) = 24
h(8) = 80
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If
A
=2
i
^
+3
j
^
+
k
^
m,
B
=
i
^
+4
j
^
−2
k
^
m, and
C
=−3
i
^
−
j
^
m, find the unknown constants a, b, and c such that a
A
+b
B
+c
C
=
k
^
. a=−11/25,b=7/25,c=1/25 a=9/25,b=−11/25,c=−1/5 a=7/25,b=−11/25,c=1/5 a=1/25,b=−7/25,c=11/25 a=11/25,b=−7/25,c=1/5
Answers
The value of a is 50, the value of b is 7, and the value of c is 1. Hence, the unknown constants a, b, and c such that aA+bB+cC=k^m are a=50, b=7, and c=1.
Given that:A = 2i^+3j^+k^m, B = i^+4j^−2k^m, and C = −3i^−j^m
We are to find the unknown constants a, b, and c such that aA+bB+cC=k^m.
So we can write the equation as (a, b, c) = k^m (2i^+3j^+k^m) + k^m (i^+4j^−2k^m) + k^m (−3i^−j^m)
Let us solve the equation (a, b, c) = k^m (2i^+3j^+k^m) + k^m (i^+4j^−2k^m) + k^m (−3i^−j^m)
Given expression can be written as (2k^m)i^ + (3k^m+4k^m-1)i^ + (-2k^m-3k^m) i^ = k^m i^ Now we know that k^m i^ = ai.
Thus, a= 2k^mIn the same way, we can get the values of b and c as follows: (3k^m+4k^m-1)j^ = bj.
Thus, b = 7k^m/25And, (-2k^m-3k^m) k^m j^ = cj^m.
Thus, c = k^m/25
Therefore, the value of a is 2k^m, the value of b is 7k^m/25, and the value of c is k^m/25.
Now, given k^m as the coefficient of the vectors, we can equate it to 1 to get the value of k^m.
Let k^m = 25. Now, putting this value in a, b, and c, we get:
a = 2k^m = 2 * 25
= 50, b
= 7k^m/25
= 7 * 25/25 = 7, and c = k^m/25
= 25/25
= 1
Therefore, the value of a is 50, the value of b is 7, and the value of c is 1.Hence, the unknown constants a, b, and c such that aA+bB+cC=k^m are a=50, b=7, and c=1.
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3 - 1/2d = 3/2d + 4 - d.
I don't get the fractions, they confuse me and I don't know what to do with them to be able to finish the problem.
Answers
The solution to the equation 3 - 1/2d = 3/2d + 4 - d is d = -7
How to evaluate the expression?The expression is given as
3 - 1/2d = 3/2d + 4 - d.
Multiply through the expression by 2
So, we have
6 - d = 3d + 8 - 2d
Evaluate the like terms
6 - d = d + 8
Evaluate the like terms
2d = -14
Divide by 2
d = -7
Hence, the solution to the equation 3 - 1/2d = 3/2d + 4 - d is d = -7
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Find the correct values for the variables that make the statement.
Answers
Answer:
hey:)
H= 90+33=123
180-123=57
so:
H=57°
abc is a right triangle with ab=ac. bisector of <a meets bc at d. prove that bc = 2ad.
Answers
Answer:
Let ac=ab=5
With this, bc= 5√2
Step-by-step explanation:
So to find ad, Let ad be x
5√2=(2)(x)
(5√2/2)= x
This proves that bc=2ad
Find the approximate perimeter of AABC plotted
below.
A(2,7)
C(6,7)
B(-4,-3)
Answers
Answer:
29.8 units
Step-by-step explanation:
Answer:
29.8
Step-by-step explanation:
Which expression is equivalent to 2(y + 11)
2y + 22
2y + 11
y + 22
Answers
Reason because you distribute the 2 to both of them
f(x)=x+5
g(x)=\sqrt 3x+1
find f/g and f-g then give their domains using interval notation
Answers
\( \frac{f(x)}{g(x)} = \frac{x + 5}{ \sqrt{3x + 1} } \)
\(x ∈ \: \: ] \frac{ - 1}{3} ,∞[\)
\(f(x) - g(x) = x + 5 - \sqrt{3x + 1} \)
\(x ∈ \: \: [ \frac{ - 1}{3} ,∞[\)
Which algebraic expression has a term with a coefficient of 9?
• A. 6 + X- 9
• B. 6x - 9
C. 6(x + 5)
• D. 9x ÷ 6
Answers
Answer:
D
Step-by-step explanation:
Coefficient is the number next to the variable, and D is the only one where that number is 9. All the others are 6.
n C (x) = 0.6x² − 168x+17,507. What is the minimum unit cost?
-
Answers
Step-by-step explanation:
Answer:
Minimum Unit Cost = $14,362
Step-by-step explanation:
The standard form of a quadratic is given by:
ax^2 + bx + c
So for our function, we can say,
a = 0.6
b = -108
c = 19,222
We can find the vertex (x-coordinate where minimum value occurs) by the formula -b/2a
So,
-(-108)/2(0.6) = 108/1.2 = 90
Plugging this value into original function would give us the minimum (unit cost):
During the Martinez family trip, they drove from home to a ski lodge, a distance of 52 4 5 miles, and then drove an additional 14 4 5 miles to visit friends.
Answers
Answer:
\(134 \frac{6}{5}\)
Step-by-step explanation:
The computation of the distance travelled is shown below:
Total distance would be
\(= 52 \frac{4}{ 5} + 14 \frac{ 4}{5}\\\\= \frac{264}{5} + \frac{74}{5}\\\\= \frac{338}{5}\)
Now if the same route would taken again in order to back home
So, the total distance would be doubled
\(= \frac{338}{5} \times 2\\\\ = \frac{676}{5} \\\\= 134 \frac{6}{5}\)
Solve for x, given that AB ≅ BC.
Answers
\(▪▪▪▪▪▪▪▪▪▪▪▪▪ {\huge\mathfrak{Answer}}▪▪▪▪▪▪▪▪▪▪▪▪▪▪\)
The value of x is ~
\( \boxed{5}\)
\( \large \boxed{ \mathfrak{Step\:\: By\:\:Step\:\:Explanation}}\)
As it's given that AB = BC, so let's plug the value of AB and BC in terms of x and equate them to solve for x ~
that is ~
\(2x - 1 = x + 4\)\(2x - x = 4 + 1\)\(x = 5 \: \: units\)Answer:
See below for answer.
Step-by-step explanation:
AB ≅ BC Given
2x - 1 = x + 4 Definition of congruent
-x = -x
x - 1 = 4 Add -x to both side of the equation
+1 = +1 Add 1 to both sides of the equation
x = 5
What is the relationship between
Answers
Answer:
betweeeeeeen
Step-by-step explanation:
which is the best estimate for the average rate of change for the quadratic function graph on the interval \(0 \leqslant x \leqslant 4\)
Answers
The average rate of change of the given quadratic function on the interval
\(0\le x\le4\)is the slope of the secant line connecting the points
\((0,f(0))\text{ and (4,f(4)}\)In other words, the average rate of change is
\(m=\frac{f(4)-f(0)}{4-0}\)From the graph, we can see that f(0)=0 and f(4)=-4. By substituying these values into the last equation, we obtain
\(\begin{gathered} m=\frac{-4-0}{4-0} \\ m=-\frac{4}{4} \\ m=-1 \end{gathered}\)Hence the average rate of change for the given quadratic function whose graph is shown on 0≤x≤4 is -1
Work out the area of the parallelogram.
Answers
Answer:
Step-by-step explanation:
it does not show the problem
the age of Edna, Ellie, and Elsa are consecutive integers. the sum of their ages is 117 what are their ages?
Answers
Answer:
Their ages are 38,39 and 40.
Step by step explanation:
If they were all the same age, then their ages would be:
\(\frac{117}{3}=39\)But, they are consecutive integers. So, you can make Elsa 40 and Edna 38.
\(38+39+40=117\)In circle P, PQ = 3 and m/QPR = 45°. Find the area of shaded sector. Express
your answer as a fraction times πT.
Answer
Answers
The calculated value of the area of shaded sector is 1.125π
Finding the area of shaded sector.From the question, we have the following parameters that can be used in our computation:
Radius, PQ = 3 units
Central angle, QPR = 45 degrees
The area of shaded sector is calculated as
Area = Central angle/360 * π * Radius^2
Substitute the known values in the above equation, so, we have the following representation
Area = 45/360 * π * 3^2
Evaluate
Area = 1.125π
Hence, the area is 1.125π
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how to solve -2(x+3)=8
Answers
Multiply -2 with x and 3
-2x -6 = 8
Now add 6 to both sides
-2x -6 = 8
+6 +8
-2x = 16
Lastly divide -2 from both sides to find x
-2x = 16
—— —-
-2x -2x
X = -8
Answer:
x=-7
Step-by-step explanation:
-2(x+3)=8
-2x-6=8
-2x=8+6
-2x=14 -2x/-2=14/-2
x= -7
if you want to be sure do the following
-2(-7+3)=8
-2(-4)=8
8=8