8.5 Exercises
In Exercises 1-20 use the Laplace transform to solve the initial value problem. Where indicated by C/G, graph the solution.
#17.
y" + 3y' + 2y = {
e⁻ᵗ, 0 ≤ t < 1
0, t ≥ 1
y(0) = 1
y'(0) = - 1
Answers
We can express the forcing function (the piecewise expression on the right side) in terms of the step function as [tex]e^{-t}(u(t) - u(t-1))[/tex] where
[tex]u(t) = \begin{cases}1&\text{for }t\ge0\\0&\text{otherwise}\end{cases}[/tex]
Let F(s) be the Laplace transform of a function f(t). Now recall the transform pair
[tex]f(t-c) u(t-c) \mapsto e^{-cs} F(s)[/tex]
This means
[tex]e^{-t} u(t) \mapsto \dfrac1{s+1}[/tex]
[tex]e^{-t} u(t-1) = \dfrac1e \times e^{-(t-1)} u(t-1) \mapsto \dfrac{e^{-(s+1)}}{s+1}[/tex]
I assume you're familiar with the transform rule for derivatives of y(t). Now we're ready to take the transform of both sides of the ODE:
[tex]y'' + 3y' + 2y = e^{-t}(u(t) - u(t-1))[/tex]
[tex]\implies \left(s^2 Y(s) - s y(0) - y'(0)\right) + 3 \left(s Y(s) - y(0)\right) + 2 Y(s) = \dfrac{1 - e^{-(s+1)}}{s+1}[/tex]
Plug in the initial values and solve for Y(s) :
[tex]\left(s^2 Y(s) - s + 1\right) + 3 \left(s Y(s) + 1\right) + 2 Y(s) = \dfrac{1 - e^{-(s+1)}}{s+1}[/tex]
[tex](s^2 + 3s + 2) Y(s) - s + 4 = \dfrac{1 - e^{-(s+1)}}{s+1}[/tex]
[tex]Y(s) = \dfrac{1 - e^{-(s+1)} + (s-4)(s+1)}{(s+1)(s^2 + 3s + 2)}[/tex]
[tex]Y(s) = \dfrac{1 - e^{-(s+1)} + (s-4)(s+1)}{(s+1)^2 (s+2)}[/tex]
Consider the partial fraction expansion
[tex]\dfrac1{(s+1)^2(s+2)} = \dfrac a{s+1} + \dfrac b{(s+1)^2} + \dfrac c{s+2}[/tex]
Solve for the coefficients:
[tex]1 = a(s+1)(s+2) + b(s+2) + c(s+1)^2[/tex]
[tex]s = -1 \implies b = 1[/tex]
[tex]s = -2 \implies c = 1[/tex]
[tex]1 = (a+c)s^2 + \cdots \implies a+c = 0 \implies a = -1[/tex]
Hence we can expand Y(s) as
[tex]Y(s) = \dfrac1{(s+1)^2} + \dfrac1{s+2} + \dfrac{e^{-(s+1)}}{s+1} - \dfrac{e^{-(s+1)}}{(s+1)^2} - \dfrac{e \times e^{-(s+2)}}{s+2}[/tex]
The last transform pair we need is
[tex]e^{ct} f(t) \mapsto F(s - c)[/tex]
Now, taking inverse transforms of everything yields
[tex]\dfrac1{(s+1)^2} \mapsto te^{-t}[/tex]
[tex]\dfrac1{s+2} \mapsto e^{-2t}[/tex]
[tex]\dfrac{e^{-(s+1)}}{s+1} \mapsto e^{-t} u(t-1)[/tex]
[tex]\dfrac{e^{-(s+1)}}{(s+1)^2} \mapsto e^{-t} (t-1) u(t-1)[/tex]
[tex]\dfrac{e \times e^{-(s+2)}}{s+2} \mapsto e^{-(2t-1)} u(t-1)[/tex]
and putting everything together gives the same solution as the one provided.
Related Questions
An evangelist traveled 748 miles in 12 hours
to get to his next meeting. What was his average rate of speed? (Write answer as a mixed number.)
pls and thank you :)
Answers
Answer:
23
Step-by-step explanation:
this is because it is 23
Answer:
speed=distance ÷time
speed=748÷12
speed=6.333
Pls help me
Pls show your answer pls and thank you
Answers
Answer: Option C
Step-by-step explanation:
I WILL GIVE 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT NOOOO SCAMS AND EXPLAIN WHY THAT IS THE ANSWER
Answers
Answer:
---------------HeyA AtorO ----------------------
[tex] \large{ \tt{ \star \: S \: O \: L \: U \: T \: I \: O \: N : }}[/tex]
We're provided a parallelogram ABCD. You must have known that the opposite angles of parallelogram are always equal.∠C = ∠A = 123° & ∠C = ∠D = x ° { suppose }
Set up an equation & solve for x :[tex] \tt{ \star \: {123 \degree + 123 \degree + x + x = 360 \degree}}[/tex]
[tex] \longrightarrow \tt \: 246 + 2x = 360[/tex]
[tex] \longrightarrow \tt \: 2x = 360 - 246[/tex]
[tex] \longrightarrow \tt 2x = 114[/tex]
[tex] \longrightarrow \tt \: x = \boxed{ \tt{57}}[/tex]
Hence , Our final answer is 57° .----------- HappY LearninG <3 ------------------
Select the correct answer.
Which statement is true about this equation?
A.
It represents neither a relation nor a function.
B.
It represents a function only.
C.
It represents both a relation and a function.
D.
It represents a relation only.
Answers
Answer:
Once the equation is given, the question can be answered, with no equation, no one can find the answer
Step-by-step explanation: I will edit my answer when the equation is given
Answer: a
Step-by-step explanation:
I hope this I better than the blurry photo I took '-'
Answers
Answer:The fourth option
decreases by 1 minute every 2 hours trained
Step-by-step explanation:
If y is how much it decreses its negative, and if the slope is -1/2 then they decrease by one for every 2 hours since the two represents x.
The radius of the sphere is 10 units. What is the approximate volume of the sphere? Use 3.14 round to nearest whole cubic unit
Answers
Answer:
2,355 cubic units
Step-by-step explanation:
We will see that the volume of the sphere is 2,355 cubic units.
How to get the volume of a sphere?
We know that for a sphere of radius R, the volume is given by:
Here we know that:
R = 10 units
pi = 3.14
Replacing that in the volume equation we get:
So the volume of the sphere is 2,355 cubic units.
Can yall help me please
Answers
Answer:
Im not good at math just breath and dont psnik
Step-by-step explanation:
Solve x2 − 20 = 0. If necessary, round to the nearest hundredth.
±4.47
400
±40
There is no solution as you cannot take the square root of a negative number.
Answers
Answer:
Option A, [tex]x =[/tex] ± [tex]4.47[/tex]
Step-by-step explanation:
Step 1: Add 20 to both sides
[tex]x^2 - 20 = 0[/tex]
[tex]x^2 - 20 + 20 = 0 + 20[/tex]
[tex]x^2 = 20[/tex]
Step 2: Square root both sides
[tex]\sqrt{x^2} = \sqrt{20}[/tex]
[tex]x =[/tex] ± [tex]4.47[/tex]
Answer: Option A, [tex]x =[/tex] ± [tex]4.47[/tex]
Is 2.89 rational or irrational
Answers
Answer:
Rational
Step-by-step explanation:
Rational numbers:
Rational numbers are such numbers that can be expressed in the form p/q, where the value of q must not be equivalent to 0.
Irrational numbers:
These numbers cannot be expressed in the form p/q, where the value of q must not be equivalent to 0.
[tex]\implies 2.89 = \dfrac{289}{100} \ \ \text{[In} \ \frac{\text{p}}{\text{q}} \ \text{form (\text{q}} \neq 0)]}[/tex]
Thus, 2.89 is a rational number.
find the value of each variable. 3x 2y 34
Answers
X= 30
Y=28
Step-by-step explanation:
It's a right-angle triangle. So, 3 x 30 will give you a 90 degree angle.
180-90=90
That means the next 2 angles should add to 90.
2x28=56
56+34=90
Hope this helps!
What is the area, in square units, of a rectangle XWYZ, with vertices X(0, 0) W(0,7) Y (3,7) Z(3,0)?
Answers
Answer:
21
Step-by-step explanation:
A=LxW=7x3=21 The length was 7 and the width was 3.
Please help! i’m horrible at math and i don’t want to fail!!! 6-1: MathXL for School: Practice & Problem Solving
Due 04/27/
5.1.PS-13
Question Help
A bag contains 9 white marbles and 54 red marbles. If a representative sample contains 2 white
marbles, then how many red marbles would you expect it to contain? Explain.
Because the ratio of white marbles to red marbles is in the population, the expected number of
red marbles is times the number of white marbles in the representative sample. There should be
red marbles.
Answers
Answer:
12 red marbles
Step-by-step explanation:
9/4.5 to get 2 white marbles
54/4.5 to get 12 white marbles
I need help with a question.
Some students are playing a game. they roll a number cube and spin the arrow on a spinner on each turn.
-The number cube has sides numbered 1 through 6.
-The spinner has 3 equal-sized sections colored red(r), yellow(y), and blue(b).
This tree diagram shows the sample space for the possible outcomes of rolling a number cube one time and then spinning the arrow on the spinner once.(I'll put the diagram on the bottom)
What is the probability that, on a student's turn, the number cube will land with an even number on the top face and the arrow on the spinner will stop on the blue(b) section?
A. 1/3
B. 3/6
C. 3/18
D. 9/16
Answers
Answer:
option c
Step-by-step explanation:
3/6probability of number cube lands on an even number
1/3 probability that it will land on the spinner
multiply 1/3 x 3/6 =3/18
let me know if i get it correct
The probability that on a student's turn, the number cube will land with an even number on the top face and the arrow on the spinner will stop on the blue(b) section is P = 3/18
What is Probability?The probability that an event will occur is measured by the ratio of favorable examples to the total number of situations possible
Probability = number of desirable outcomes / total number of possible outcomes
The value of probability lies between 0 and 1
Given data ,
The probability of the number cube landing with an even number on the top face is 3/6 or 1/2 since there are three even numbers (2, 4, and 6) out of the six possible outcomes (1, 2, 3, 4, 5, and 6)
The probability of the arrow on the spinner stopping on the blue section is 1/3 since there is only one blue section out of the three possible outcomes
To find the probability of both events occurring together, we multiply the probabilities of each event:
P(even number and blue section) = P(even number) × P(blue section)
P(even number and blue section) = 1/2 × 1/3
P(even number and blue section) = 1/6
P = 3/18
Therefore, the probability that, on a student's turn, the number cube will land with an even number on the top face and the arrow on the spinner will stop on the blue section is 1/6.
Hence , the probability is P = 3/18
To learn more about probability click :
https://brainly.com/question/17089724
#SPJ2
A skateboard is marked 20% off. If the board originally cost $45.00, what is the sale price?
Answers
Answer:
$36.00
Step-by-step explanation:
original cost= $45.00
discount= 20%
to find:the sale price of the skateboard.
solution:[tex]original \: price \times \frac{discount \%}{100} [/tex]
[tex] = 45.00 \times \frac{20}{100} [/tex]
[tex] = 45 \times 0.2[/tex]
[tex] = 9.00[/tex]
[tex]45 - 9[/tex]
[tex] = 36.00[/tex]
therefore, the sale price of the skateboard is $36.00
PLEASE HELP WILL MARK BRAINLIEST THANK YOU
Answers
Answer: x = -6
Step-by-step explanation:
The line of x = -6 will be parallel to the y-axis because it is a vertical line, and the y-axis runs vertically.
I have graphed the line of x = -6 so you can see it and compare it with the y-axis. See attached.
A 400 pound metal star is hanging on two cables which are attached to the ceiling. The left hand cable makes a 18° angle with the ceiling while the right hand cable makes a 72° angle with the ceiling. What is the tension on each of the cables? (Round your answers to three decimal places.)
Answers
Answer:
left: 123.607 lbright: 380.423 lbStep-by-step explanation:
By balancing horizontal and vertical forces, we find the cable tensions to be ...
Ta = W·sin(b)/sin(a+b) . . . . . where W is the weight being held
Tb = W·sin(a)/sin(a+b)
Where Ta is the tension in the cable that makes an angle of 'a' with respect to the vertical, and Tb is the tension in the cable that makes an angle of 'b' with respect to the vertical.
__
The given angles are with respect to the ceiling, so the angles with respect to the vertical will be their compmements.
left cable (a)angle 'a' is 90° -18° = 72°
angle 'b' is 90° -72° = 18°
a+b = 72° +18° = 90°
Ta = (400 lb)sin(18°)/sin(90°) = 123.607 lb
right cable (b)Tb = (400 lb)sin(72°)/sin(90°) = 380.423 lb
_____
Additional comment
The nice expressions for cable tension come from the balance of forces.
vertical: Ta·cos(a) +Tb·cos(b) = W
horizontal: Ta·sin(a) = Tb·sin(b)
Solving the horizontal equation for Ta, we get ...
Ta = Tb·sin(b)/sin(a)
Substituting into the vertical equaiton gives ...
Tb·sin(b)cos(a)/sin(a) +Tb·cos(b) = W
Multiplying by sin(a) gives ...
Tb(sin(b)cos(a) +sin(a)cos(b)) = W·sin(a)
Using the trig identity for the sine of the sum of angles, we can rewrite this in the form shown above:
Tb = W·sin(a)/sin(a+b)
The problem is symmetrical with respect to 'a' and 'b', so the other tension is found by interchanging 'a' and 'b' in the equation:
Ta = W·sin(b)/sin(a+b)
Part A)
Please solve this system of equations using the elimination method ONLY.
3x - 2t = 4
2x - 5 = -t
Part B)
Which number is the most convenient to multiply by when solving this linear system of equations by elimination?
A) -3
B) -1
C) 1
D) 2
Thank you in advance.
Answers
Answer:
D) 2
Step-by-step explanation:
When solving a system of equations using elimination, it often works well to start with them in a form that has all of the variables on the same side of the equal sign. Putting the second equation into standard form will accomplish that:
3x -2t = 42x +t = 5To eliminate one of the variables from the system, the equations are multiplied by numbers and combined (added or subtracted) so that the coefficient of one of the varibles becomes 0.
__
Choosing a multiplierThe coefficients of x are 3 and 2. If we are to eliminate x, we either need to multiply one equation by a rational number (2/3 or 3/2), or we need to multiply the two equations by two different numbers (2, 3).
The coefficients of y are -2 and 1. If we are to eliminate y, it is convenient to multiply the second equation by 2. This is probably the most convenient choice.
__
SolutionMultiplying the second equation by 2 and adding it to the first equation, we have ...
(3x -2t) +2(2x +t) = (4) +2(5)
7x = 14 . . . . . simplify; the y-variable is eliminated
x = 2 . . . . . . divide by 7
Using the original second equation, we have ...
2(2) -5 = -t = -1
t = 1
So, the solution is (x, t) = (2, 1).
Kyle needs 4 feet of ribbon for an art project. He has 20 inches of ribbon. How many
more inches does Kyle need for the project?
1 foot = 12 inches
Answers
Answer:
28 Inches more
Step-by-step explanation:
1 ft = 12 in
He needs 4ft
12x4=48
So he needs 48 inches
He has 20 inches
48-20=28
He needs 28 inches
A golfer hits a ball from an elevated tee 75 feet high at an initial velocity of 115 ft/sec at angle of elevation of 57⁰ a. Write a parametric equation that describes the position of the ball at time t.
Answers
x=115tcos57
y=115tsin57-16t^2
PLEASE HELP WILL MARK BRAINLIEST THANK YOU
Answers
Answer:
When used as a diacritic mark, the term dot is usually reserved for the interpunct, or to the glyphs "combining dot above" and "combining dot below" which may be combined with some letters of the extended Latin alphabets in use in Central European languages and Vietnamese
Given the diagram below, if CD = x + 4,
DE = 2x + 7, and CE = 7x-25, what is the value
of x?
Answers
9
Explanation :
check out my attached work
Hope it helps, let me know if you have any questions !
Have a nice rest of your day :)
For what value of x does 32X = 93x-4?
=
1
2
3
4
Answers
Answer:
B. 2
Step-by-step explanation:
So when we are dealing with exponents identify the following terms
Base: the number being multiplied by itself
Exponent: the number of times you multiply the base number by itself
So in this problem we would have to find what x is in order to find our exponents, you can do this by substiting each number for x and solving as is
[tex]3^4=3*3*3*3=81[/tex]
[tex]9^2=9*9=81[/tex]
can someone help me pls
Answers
Part A
Multiply each age with its corresponding frequency
19*2 = 3820*3 = 6021*1 = 2122*4 = 8823*1 = 23Then add up those products
38+60+21+88+23 = 230
We divide that sum over the total frequency of 11 (since 2+3+1+4+1 = 11).
mean = 230/11 = 20.9 approximately
Answer: 20.9Nice work on getting the correct answer.
============================================================
Part B
x = age of the new player
Adding x to the sum of the products 230, and then dividing over 11+1 = 12 will get us the new mean of 22.
The 230 represents the sum of the ages of the original eleven people. The expression 230+x represents the sum of the twelve ages.
So,
(230+x)/12 = 22
230+x = 12*22
230+x = 264
x = 264-230
x = 34
Answer: 34Describe the pattern
Answers
Answer:
negative linear
Step-by-step explanation:
as you see the correlation is going negative in a linear line.
I NEED HELP ASAP! PLEASE LET ME KNOW WHICH ONE
Answers
Step-by-step explanation:
3 parts in figure
1 cube and 2 hemispheres
surface area of cube = 6a^2= 6*144= 864 mm^2
surface area of hemisphere = 3*π*r^2 =108π
2 hemispheres = 214π mm^2
total surfaCe area = 1542 mm^2
se the number line to answer the question.
A number line has end points marked at 5 tenths and 6 tenths with 9 tick marks between. Point A is labeled on the third mark after 5 tenths.
Select all of the following that give the value of point A.
A.
53
100
B.
5
3
100
C.
53
10
D.
0.53
E.
5.3
Answers
Answer:
[tex]\sf 0.53\quad and \quad \dfrac{53}{100}[/tex]
Step-by-step explanation:
Convert the end points to decimals:
[tex]\textsf{5 tenths}=\dfrac{5}{10}=5 \div 10=0.5[/tex]
[tex]\textsf{6 tenths}=\dfrac{6}{10}=6\div 10=0.6[/tex]
If the number line has 9 tick marks between the endpoints, there will be 10 equal spaces (see attachment), so the tick marks denote hundredths.
The difference between 0.6 and 0.5 = 0.1
Divide this by 10: 0.1 ÷ 10 = 0.01
Therefore, each tick mark on the number line denotes hundredths and is 0.01 greater than the previous tick mark.
If Point A is labeled on the third mark after 5 tenths then:
A = 0.5 + 0.01 + 0.01 + 0.01 = 0.53
0.53 as a fraction:
[tex]\implies 0.53 = \dfrac{0.53}{1}=\dfrac{0.53 \times 100}{1 \times 100}=\dfrac{53}{100}[/tex]
Therefore, point A is:
[tex]\sf 0.53\quad and \quad \dfrac{53}{100}[/tex]
Answer: i need points
Step-by-step explanation:
Mr Ahmad bought 2 fans and 3 table lamps for $540. Each fan cost 3 times as much as a table lamp. (a) Find the cost of each table lamp. (b) Find the cost of each fan.
Answers
Answer:
(a) $60
(b) $180
Step-by-step explanation:
Let the price of 1 table lamp be t.
Then the price of a fan is 3t.
2(3t) + 3t = 540
6t + 3t = 540
9t = 540
t = 60
A table lamp costs $60.
3($60) = $180
A fan costs $180.
Bowling alley charges a fee for every game bowled plus initial fee for shoe rental.
Answers
Answer:
Whats the numbers can u give me them?
Step-by-step explanation:
PLEASE HELP ME WITH THIS!!!
Answers
Answer:
a) 70°
Step-by-step explanation:
∠3 = 110
∠5 = ∠1
∠1 = 180 - ∠3
∠1 = 180 - 110
∠ 1 = 70
∠5 = ∠1 (corresponding angles postulate)
∠5 = 70
What is the range of the relation below?
Answers
Answer:
a
Step-by-step explanation:
you can see on y in which range the line goes, I hope its correct
Answer:
Step-by-step explanation:
None of them its:
-3.2 ≤ y ≤ 4.