3c. For each differential equation, find the Laplace transform of the solution y:
y'' + 3y' + 2y = 5 sin(πx), y(0) = y'(0) = 0.
Answers
Answer:
[tex]y(x)=\frac{5}{3\pi}-\frac{5\pi}{27+3\pi^2}e^{-3x}-\frac{15}{9\pi+\pi^3}cos(\pi x)-\frac{5}{9+\pi^2}sin(\pi x)[/tex]
Step-by-step explanation:
[tex]y''+3y'+2y=5sin(\pi x),\: y(0)=y'(0)=0\\\\\mathcal{L}\{y''\}+3\mathcal{L}\{y'\}+2\mathcal{L}\{y\}=\mathcal{L}\{5sin(\pi x)\}\\\\s^2Y(s)-sy(0)-y'(0)+3[sY(s)-y(0)]+2Y(s)=\frac{5\pi}{s^2+\pi^2}\\\\s^2Y(s)+3sY(s)=\frac{5\pi}{s^2+\pi^2}\\ \\(s^2+3s)Y(s)=\frac{5\pi}{s^2+\pi^2}\\ \\Y(s)=\frac{5\pi}{(s^2+\pi^2)(s^2+3s)}\\ \\Y(s)=\frac{5\pi}{s(s+3)(s^2+\pi^2)}[/tex]
Perform the partial fraction decomposition
[tex]\frac{5 \pi}{s \left(s + 3\right) \left(s^{2} + \pi^{2}\right)}=\frac{A}{s}+\frac{B}{s + 3}+\frac{C s + D}{s^{2} + \pi^{2}}\\5\pi=s \left(s + 3\right) \left(C s + D\right) + s \left(s^{2} + \pi^{2}\right) B + \left(s + 3\right) \left(s^{2} + \pi^{2}\right) A\\\\5 \pi=s^{3} A + s^{3} B + s^{3} C + 3 s^{2} A + 3 s^{2} C + s^{2} D + \pi^{2} s A + \pi^{2} s B + 3 s D + 3 \pi^{2} A\\\\5\pi=s^3(A+B+C)+s^2(3A+3C+D)+s(\pi^2A+\pi^2B+3D)+3\pi^2A[/tex]
Solve for each constant
[tex]\begin{cases} A + B + C = 0\\3 A + 3 C + D = 0\\\pi^{2} A + \pi^{2} B + 3 D = 0\\3 \pi^{2} A = 5 \pi \end{cases}[/tex]
[tex]3\pi^2A=5\pi\\A=\frac{5}{3\pi}[/tex]
[tex]A+B+C=0\\3A+3B+3C=0\\(3A+3B+3C=0)-(3A+3C+D=0)\\3B-D=0\\3B=D[/tex]
[tex]\pi^2A+\pi^2B+3D=0\\\pi^2(\frac{5}{3\pi})+\pi^2B+3(3B)=0\\\frac{5\pi}{3}+\pi^2B+9B=0\\ B(\pi^2+9)=-\frac{5\pi}{3}\\B=-\frac{5\pi}{3(\pi^2+9)}\\B=-\frac{5\pi}{3\pi^2+27}[/tex]
[tex]3B=D\\3(-\frac{5\pi}{3\pi^2+27})=D\\-\frac{5\pi}{\pi^2+9}=D[/tex]
[tex]A+B+C=0\\\frac{5}{3\pi}-\frac{5\pi}{3\pi^2+27}+C=0\\\frac{15\pi^2+135}{3\pi(3\pi^2+27)}-\frac{15\pi^2}{(3\pi)(3\pi^2+27)}+C=0\\\frac{135}{9\pi^3+81\pi}+C=0\\\frac{15}{\pi^3+9\pi}+C=0\\C=-\frac{15}{\pi^3+9\pi}[/tex]
Take the inverse transform and solve for the IVP
[tex]Y(s)=\frac{\frac{5}{3 \pi}}{s}+\frac{- \frac{5 \pi}{27 + 3 \pi^{2}}}{s + 3}+\frac{- \frac{15 s}{9 \pi + \pi^{3}} - \frac{5 \pi}{9 + \pi^{2}}}{s^{2} + \pi^{2}}\\\\Y(s)=\frac{\frac{5}{3 \pi}}{s}-\frac{\frac{5 \pi}{27 + 3 \pi^{2}}}{s + 3}-\frac{ \frac{15 s}{9 \pi + \pi^{3}}}{s^{2} + \pi^{2}}-\frac{\frac{5\pi}{9+\pi^2} }{s^2+\pi^2}\\ \\y(x)=\frac{5}{3\pi}-\frac{5\pi}{27+3\pi^2}e^{-3x}-\frac{15}{9\pi+\pi^3}cos(\pi x)-\frac{5}{9+\pi^2}sin(\pi x)[/tex]
Related Questions
1 Josiah is making a candle by pouring melted wax into a mold in the shape of a square pyramid. Each side of the base of the pyramid is 12 in and the height of the pyramid is 14in. To get the wax for the candle, Josiah melts cubes of wax that are each 6 in by 6 in by 6 in. How many of the wax cubes will Josiah need in order to make the candle? Show your work.
Answers
Answer:
3.11 wax cubes
Step-by-step explanation:
Which sequence of steps is a correct derivation of the difference quotient for f(x) = ln(4 – x) – 1?
Answers
Answer:
Step-by-step explanation:
What is the Square root of 3664?
Answers
Answer:
60.5
Thanks!
Mark me brainliest!
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Dudley Durite deposited $350 into a savings account that earns 6.5% interest compounded semi annually. What amount was in his account at the end of 1 year?
Answers
Answer:
$373.12
Step-by-step explanation:
You want to set this up with the equation for compound interest which is [tex]A=P(1+\frac{r}{n})}^{nt}[/tex]. A stands for amount, P stands for principle which in this case is 350, r is rate which for this one is .065, ad n is the amount of times compounded per year, and since it is compounded semi annually you do 2 for 2 times a year. Your equation based on this information should look like this:
[tex]A=350(1+\frac{.065}{2})}^{(2)(1)}[/tex]
After working this out you get A = 373.12
$373.12
Use Heron's formula to find the area of the triangle. Round to the nearest square unit.
a = 8 inches, b = 13 inches, c = 7 inches
Group of answer choices
39 square inches
23 square inches
21 square inches
41 square inches
Answers
Answer:
24.25 in²
Step-by-step explanation:
s = (a + b + c)/2 = (8 + 13 + 7)/2 = 14 in
s - a = 6, s - b = 1, s - c = 7
so
the area = [tex]\sqrt{14\times6\times1\times7}=24.25[/tex]
10 yd
10 yd
Find the area of each?
Answers
Answer:
20 yards
Step-by-step explanation:
i got 100 0n the quiz
Describe what happens if you divide a difference by 3.
Answers
find the value of x ?
Answers
Answer:
Step-by-step explanation:
∠E = ∠G
∠D= ∠F because it’s a parallelogram ==> D=F=2x+12 and E=G=5x
5x+2x+12=180
7x=168
X=24
Solution 2:
2*5x+2*(2x+12)=360
10x+4x+24=360
14x=360-24=336
X=336/14
X=24
Answer:
[tex]{\fbox{\fbox{\fbox{\fbox{\fbox{\fbox{\fbox{\fbox{\fbox{\fbox{\fbox{\fbox{☺☺Hope\ it\ helps\ you}}}}}}}}}}}}}[/tex]
A/Q
5x+2x+12=180(Sum of adjacent angles is 180)
=>7x+12=180
=>7x=180-12
=>x=168/7
Hence,x=24....
Use the square root property to solve the equation.
x^2=153
Answers
Answer:
3√(17) or 12.3693
Step-by-step explanation:
x^2 = 153
take square root from both side...
√(x^2) = √(153)
then...
x=√(153)
reduce to get...
x=3√(17)
the approximation is...
12.3693
Please rate!!
Answer:
x = 3sqroot17
x ~= 12.37
Step-by-step explanation:
To use the square root property, take the square root of both sides of the equation. see image.
Sqroot17 can be simplified in radical form or a decimal approximation.
For the radical form note that 153 is 9×17.
Since 9 is a perfect square, it simplifies a little bit. see image.
Which graph represents the function?
f(x)={3 if x<1x if x≥1
Answers
Answer:
Top Right
Step-by-step explanation:
If it's x≥1, you would plot a closed (or filled) circle at 1, and it would continue going right. If it's x<1, you would plot an open (or an o) circle with a line continuing left.
The function represents f(x)={3 if x<1x if x≥1 in Figure 4 is correct.
What is slope?In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction. There is no clear reason why the letter m is employed for slope, however its earliest use in English can be found in O'Brien (1844), who wrote the equation of the a straight line as "y = mx + b," and it can also found in Todhunter (1888), who wrote it as "y = mx + c." Slope is frequently denoted by the letter x. The ratio of the "vertical change" to the "horizontal change" between (any) two distinct points along a line is used to calculate slope. The ratio can also be written as a quotient ("rise over run"), which produces the same number for each and every two distinct points on a single line.
Here,
For x≥1 ⇒ f(x)=x
For x<1 ⇒ f(x)=3
Both these statements are followed together only in figure 4.
Hence, figure 4 is correct.
Learn more about slope from the link.
https://brainly.com/question/3493733
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Divide. Write your answer in simplest form.
8/7÷4
Answers
Answer:
8/7 ÷ 4/1 = 7/32
Step-by-step explanation:
4/1 is the same as 4, flip all the Fractions so the bottom number is on the top and top on bottom and then multiply top x top and bottom x bottom
Hey there!
First, let's convert 4 into a fraction:
4=4/1
Now, divide the fractions:
[tex]\bold{\displaystyle\frac{8}{7} :\frac{4}{1}}[/tex]
Now, flop the second fraction over and multiply the first fraction times the second one:
[tex]\bold{\displaystyle\frac{8}{7} *\frac{1}{4}}[/tex]
Multiply the numerator (8) times the numerator of the second fraction (1) and multiply the denominator (7) times the denominator of the second fraction (4):
[tex]\bold{\displaystyle\frac{8}{28}}}[/tex]
Also, this fraction can be reduced:
[tex]\bold{\displaystyle\frac{4}{14}}[/tex]
We can divide by 2 again.
Hence, the answer is
[tex]\boxed{\boxed{\bold{\displaystyle\frac{2}{7}} }}}[/tex]
Hope everything is clear.
Let me know if you have any questions!
:-)
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[tex]\rule{200}{2}[/tex]
What is the solution to the system of equations?
(4, 7)
(7, 4)
(12, 5)
(5, 12)
Answers
What is the solution to the system of equations?
3x+2y = 39
5x-y = 13
Answer ↷(5,12)
Solution ↷Given that,
3x + 2y = 39......(1)
5x - y = 13.......(2)
multiplying equation 1 by 5 & 2nd by 3 ,we get,
15x + 10y = 195.......(3)
15x - 3y = 39........(4)
Subtracting equation 4th by equation 3rd,
we get,
15x + 10y = 195
-15x - 3y = 39
_____________
ㅤ ㅤ13y = 156
ㅤㅤㅤ y = 12
_____________
so , y = 12 & x = 5
⤿5x-12 = 13⤿5x = 25⤿x = 5_____________
hence, option 4th ⇢ (5,12) is correct ✓
Graph Attached
Solution is (5,12)Option D
plz help idk what to do :(
Answers
Answer:
x=5°
∠JIK=110°
Step-by-step explanation: From question,
20x-10+8x-20=140-6x
28x-30=140-6x
34x=170
x=170/34
x=5°
Eq. in ∠JIK,
140-30
=110°
Is 2.04 is greater or less than 2.040?
Answers
Answer:
See below
Step-by-step explanation:
2.04 is neither greater nor less than 2.040
but 2.04 is equal to 2.040.
[tex]2.04=2.040[/tex]
Which of the following r-values represents the strongest correlation?
Which of the following r-values represents the strongest correlation?
A. -0.51 B.-0.71 C. -0.41 D. -0.61
Answers
Answer:
-0.71
Step-by-step explanation:
Generally, a value of r greater than 0.7 is considered a strong correlation. Anything between 0.5 and 0.7 is a moderate correlation, and anything less than 0.4 is considered a weak or no correlation.
The r-value that represents the strongest correlation is -0.71.
What is correlation?Any of the many metrics used to describe the strength and direction of a linear relationship between two random variables is called a correlation coefficient.
So, r stands for the correlation coefficient.
Depending on the direction of the association, an r-value may be either negative or positive.
Yet, the link is greater the closer an r-value is to 1.
Here - 0.71 is closer to -1 then it is having a strongest correlation.
Learn more about correlation coefficient here:
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It is found that on average, 9% of the population recycles its garbage. Under this assumption, if a random sample of 120 households in Iowa City is taken, would you be surprised to find that fewer than 3% of households in the sample recycle their garbage? Use the fact that p-hat has an approximately normal distribution. Group of answer choices I would not be surprised because there are lots of lazy students in Iowa City. I would not be surprised because 3% is not that far away from 9%. I would be surprised because Iowa City is a very clean community which means lots of people recycle their garbage. I would be surprised because if the 9% average is true, the chance of a sample proportion being less than 3% is very small (only 1%). Who cares? It's only garbage!
Answers
Complete Question
It is found that on average, 9% of the population recycles its garbage. Under this assumption, if a random sample of 120 households in Iowa City is taken, would you be surprised to find that fewer than 3% of households in the sample recycle their garbage? Use the fact that p-hat has an approximately normal distribution.
Group of answer choices
A I would not be surprised because there are lots of lazy students in Iowa City.
B I would not be surprised because 3% is not that far away from 9%.
C I would be surprised because Iowa City is a very clean community which means lots of people recycle their garbage.
D I would be surprised because if the 9% average is true, the chance of a sample proportion being less than 3% is very small (only 1%). Who cares? It's only garbage!
Answer:
The correct option is D
Step-by-step explanation:
From the question we are told that
The sample size is n = 120
The mean of the proportion is p = 0.09
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{\frac{p(1-p)}{n} }[/tex]
=> [tex]\sigma = \sqrt{\frac{0.09(1-0.09)}{120 } }[/tex]
=> [tex]\sigma = 0.02612[/tex]
Generally the chance that fewer than 3% of households in the sample recycle their garbage is mathematically represented as
[tex]P(X < 0.03) = P( \frac{X - p}{\sigma} < \frac{0.03 -0.09}{0.02612} )[/tex]
[tex]P(X < 0.03) = P( Z < -2.297 )[/tex]
From the z-table
[tex]P( Z < -2.297 ) = 0.01[/tex]
=> [tex]P(X < 0.03) = 0.01/tex]
=> [tex]P(X < 0.03) = 1\% /tex]
Label each triangle
Answers
Answer: Top right and left are ACUTE, Bottom right is RIGHT, and bottom left is OBTUSE.
Answer:
the top two are acute triangles
the third(bottom left is an obtuse angle
and the last one(bottom right) is a right triangle
Step-by-step explanation:
the top angles are all under 90 degrees which is what makes them acute
the bottom left triangle has an angle greater than 90 which makes it obtuse
and the bottom right has a right angle, which makes it a right triangle
2(3x-4)=3x+1
I just need help solving this-
Answers
Answer:
x=3
Step-by-step explanation:
simplify
6x−8=3x+1
subtract 3x
3x−8=1
add 8 to each side
3x=9
divide both sides by 3
x=3
At Skyrocket Amusement Park, 30% of the rides are roller coasters. There are a total of
18 roller coasters. How many rides are there at Skyrocket Amusement Park in all
Answers
Considering the value of the 100%
If 30% = 18 roller coasters
then 100% = 100 x 18/30
= 60 rides
Hence there are 60 rides in all
look at this equation and the steps used to solve it. Which answer is the best justification for step 1 of this solution
Answers
Answer:
equal quantities something...
Step-by-step explanation:
Where do humans and other animals get their food?
Answers
Answer:
Humans and other animals get their food from plants directly or indirectly. Humans and animals are heterotrophic organisms, so they obtain their food from other organisms.
Step-by-step explanation:
∧
Answer:
From plants and other animals
Step-by-step explanation:
Got it right on my quiz haha..
2x+3≤x−5 solve for x
Answers
Answer:
x≤−8
Step-by-step explanation:
2x+3≤x−5
Step 1: Subtract x from both sides.
2x+3−x≤x−5−x
x+3≤−5
Step 2: Subtract 3 from both sides.
x+3−3≤−5−3
Answer is
x≤−8
Which statement is true about the prime polynomial 2x2 + 3x + 3?
It cannot be modeled with a rectangle.
It can be modeled with a square.
It cannot be modeled with all positive tiles.
It can only be modeled with an odd number of tiles.
Answers
Answer:
A
Step-by-step explanation:
because problem make more sences
Answer:
A it cannot be modeled with a rectangle
Step-by-step explanation:
solve for x ~
[tex]\bold{2(x + 1) - 4x = 3x + 5} [/tex]
spammers stay away .-.
thankyou ~
Answers
Answer:
x = -0.6
solving steps:
[tex]\sf \rightarrow 2\left(x+1\right)-4x=3x+5[/tex]
remove the parenthesis
[tex]\sf \rightarrow 2x+2\right)-4x=3x+5[/tex]
group the variables
[tex]\sf \rightarrow 2x-4x-3x\right)=5-2[/tex]
simplify
[tex]\sf \rightarrow -5x\right)=3[/tex]
change sides
[tex]\sf \rightarrow x\right)=\dfrac{3}{-5}[/tex]
answer in fraction
[tex]\sf \rightarrow x\right)=-\dfrac{3}{5}[/tex]
answer in decimals
[tex]\sf \rightarrow x\right)=-0.6[/tex]
Answer:
[tex]x=-0.6[/tex]
Step-by-step explanation:
[tex]2\left(x+1\right)-4x=3x+5\quad :\quad x=-\frac{3}{5}\quad \left(\mathrm{Decimal}:\quad x=-0.6\right)[/tex]
[tex]2\left(x+1\right)-4x=3x+5[/tex]
[tex]2(x+1)-4:-2x+2[/tex]
[tex]-2x+2=3x+5[/tex]
[tex]\mathrm{Subtract\:}2\mathrm{\:from\:both\:sides}[/tex]
[tex]-2x+2-2=3x+5-2[/tex]
[tex]\mathrm{Simplify}[/tex]
[tex]-2x=3x+3[/tex]
[tex]\mathrm{Subtract\:}3x\mathrm{\:from\:both\:sides}[/tex]
[tex]-2x-3x=3x+3-3x[/tex]
[tex]\mathrm{Simplify}[/tex]
[tex]-5x=3[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}-5[/tex]
[tex]\frac{-5x}{-5}=\frac{3}{-5}[/tex]
[tex]\mathrm{Simplify}[/tex]
[tex]x=-\frac{3}{5}[/tex]
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This table shows the ratio of daisies to roses.
Complete the statement about the equivalent ratios in
this table.
Daisies
Roses
To find the values in each row,
the values
in the previous row by
256
192
64
48
16
12
Answers
Answer:
Its divide and 4
Step-by-step explanation:
To find the values in each row divide the values in previous row by 4.
What is the Ratio?
The ratio can be defined as the comparison of the fraction of one quantity towards others. e.g.- water in milk.
Clearly, from observation,
Each column represent geometric progression with common ratio r = 1/4. So when moving downward, the element in the rows decreases as multiple of 1/4.
Here,
R(1, 1) = 256
R (2 ,1) = 64
R(2,1)/R(1,1)=1/4
Thus, To find the values in each row divide the values in previous row by 4.
Learn more about Ratio here:
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GUYS HELP ME WITH THIS WILL MARK BRANLIEST
Create a system of linear equations with no solution. In your final answer, include the system of equations and the graphs of the lines.
Answers
Answer:
A border of 3 cm width is printed on 500 sheets of paper, each of length 40 cm and width 30 cm fit costs 50p to print 100 cm of border, find the total cost of printing the border.
I chose the first one please help if its not correct. Simplify.
y+ (15+ y)
15 + 2 y
17 y
15 y
15+ y
Answers
Answer:
15+2y
Step-by-step explanation:
take y to the other side
For 23-25, use the table.
In Rhonda's school, the average amount
of time spent on homework on
weeknights was 56 minutes. The table
shows how far above
and
below average
five students' times were.
Student
Difference
Fred -5
Gina 0
Hanna 3
Ilsa -3
Jon 5
23. Which students' differences were
opposite?
ANSWER ASAP PLS
Answers
Answer:
Fred and Jon
Ilsa and Hanna
Step-by-step explanation:
Fred has a difference of -5. Jon has a difference of + 5.
Positive and negative 5 are opposite.
Ilsa had a difference of -3. Hanna had a difference of + 3.
Positive and negative 3 are opposite.
a train moves at a speed of 60 km per hour.how far will it travel in 30 minutes
Answers
Answer:
The train would travel 30km in 30 minutes.
Step-by-step explanation:
Speed of train = 60km/hr
30 minutes = I hour ÷ 2
If 30 minutes is equal to half an hour, and it takes an hour for the train to travel 60 km, then you would have to half 60 km to find the length travelled in 30 minutes.
Equation: 60km ÷ 2 = 30km
D=1/3fk, solve for f
Answers
Answer:
f=3d/k
Step-by-step explanation: