1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
miss Akunina [59]
1 year ago
5

Use the laplace transform to solve the given system of differential equations. dx dt + 3x + dy dt = 1 dx dt − x + dy dt − y = et

x(0) = 0, y(0) = 0
Mathematics
1 answer:
blagie [28]1 year ago
7 0

Let X(s) and Y(s) denote the Laplace transforms of x(t) and y(t).

Taking the Laplace transform of both sides of both equations, we have

\dfrac{dx}{dt} + 3x + \dfrac{dy}{dt} = 1 \implies \left(sX(s) - x(0)\right) + 3X(s) + \left(sY(s) - y(0)\right) = \dfrac1s \\\\ \implies (s+3) X(s) + s Y(s) = \dfrac1s

\dfrac{dx}{dt} - x + \dfrac{dy}{dt} = e^t \implies \left(sX(s) - x(0)\right) - X(s) + \left(sY(s) - y(0)\right) = \dfrac1{s-1} \\\\ \implies (s-1) X(s) + s Y(s) = \dfrac1{s-1}

Eliminating Y(s), we get

\left((s+3) X(s) + s Y(s)\right) - \left((s-1) X(s) + s Y(s)\right) = \dfrac1s - \dfrac1{s-1} \\\\ \implies X(s) = \dfrac14 \left(\dfrac1s - \dfrac1{s-1}\right)

Take the inverse transform of both sides to solve for x(t).

\boxed{x(t) = \dfrac14 (1 - e^t)}

Solve for Y(s).

(s - 1) X(s) + s Y(s) = \dfrac1{s-1} \implies -\dfrac1{4s} + s Y(s) = \dfrac1{s-1} \\\\ \implies s Y(s) = \dfrac1{s-1} + \dfrac1{4s} \\\\ \implies Y(s) = \dfrac1{s(s-1)} + \dfrac1{4s^2} \\\\ \implies Y(s) = \dfrac1{s-1} - \dfrac1s + \dfrac1{4s^2}

Taking the inverse transform of both sides, we get

\boxed{y(t) = e^t - 1 + \dfrac14 t}

You might be interested in
A law firm is going to designate associates and partners to a big new case. The daily rate charged to the client for each associ
Ronch [10]

Answer:

The system of equations representing the situation are

600a+1200p = 15,000

a+p = 16

with solutions

p =9

a = 7

Step-by-step explanation:

Let us call a the number of associates and p the number of partners. If each associate costs $600 and each partner costs $1200, and it cost the law firm a total of $15,000, then we have

600a+1200p = 15,000.

Since the law firm assigned at total of 16 lawyers, we have

a+p = 16.

Thus, the system of equations representing the situation are

600a+1200p = 15,000

a+p = 16

And the solution to this system is found by solving for a in the second equation and substituting its value in the first equation:

a = 16-p

600(16-p)+1200p = 15,000

9600-600p+1200p=15,000

9600+600p=15,000

600p=5400

\boxed{ p=9}

We put this value into a+p = 16 and solve for a:

a+9=16

\boxed{a = 7}

4 0
3 years ago
F(x) = — 3х2 – 6х +5 <br> maximum and minimum value
gregori [183]

9514 1404 393

Answer:

  maximum: 8; no minimum

Step-by-step explanation:

A graph can be useful. I find a graphing calculator handy. It shows the maximum of the function is f(-1) = 8. Since the parabola goes to -∞ for large values of x, there is no minimum.

  maximum: 8

__

You can also find the maximum by putting the function in vertex form.

  -3(x^2 +2x) +5 . . . . factor the leading coefficient from the x terms

 -3(x^2 +2x +1) +5 -(-3)(1) . . . . add the square of half the x-coefficient, subtract the equivalent amount

  -3(x +1)^2 +8 . . . . . . the vertex form of the expression for f(x)

This form is ...

  a(x -h)^2 +k . . . . . with a=-3, h=-1, k=8

so the vertex is (h, k) = (-1, 8) -- the same as shown on the graph. The negative value of 'a' tells you the parabola opens downward, so the vertex is the maximum. The maximum is 8 at x = -1.

5 0
3 years ago
Which unit rate is the lowest price per ounce? (5 points)
SVETLANKA909090 [29]

Answer:choice b

Step-by-step explanation:

I just took the trst

6 0
3 years ago
Mrs. Daly's reading classes were surveyed about their reading preferences. The data is summarized in the table below.
andriy [413]

Answer:

Required probability is 0.19

Step-by-step explanation:

The two-way table for the given data is,

                              Fiction                Non-Fiction                Total

9th grade                35                            8                             43

10th grade               52                            12                           64

Total                        87                            20                           107

It is required to find the probability of the students who prefer to read non-fiction novels, given that they are in 9th grade.

That is, to find the conditional probability P(Non-fiction| 9th\ grade).

Now, P(A|B)=\dfrac{P(A\bigcap B)}{P(B)}

So, we get,

P(Non-fiction| 9th\ grade)=\dfrac{(Non-fiction\bigcap 9th\ grade)}{P(9th\ grade)}\\\\P(Non-fiction| 9th\ grade)=\dfrac{8}{43}\\\\P(Non-fiction| 9th\ grade)=0.19

<h3>Thus, the required probability is 0.19.</h3>
5 0
3 years ago
A plastic box has a length 4 cm longer than its width. The area of the rectangular top of the box is 77 square cm. Find the leng
finlep [7]

Answer: the length is 11 cm.

The width is 7 cm.

Step-by-step explanation:

Let L represent the length of the rectangular plastic box.

Let W represent the width of the rectangular plastic box.

The area of the rectangular top of the box is 77 square cm. This means that

LW = 77- - - - - - - ;- - - -1

The plastic box has a length 4 cm longer than its width. This means that

L = W + 4

Substituting L = W + 4 into equation 1, it becomes

(W + 4)W = 77

W² + 4W = 77

W² + 4W - 77 = 0

W² + 11W - 7W - 77 = 0

W(W + 11) - 7(W + 11) = 0

W - 7 = 0 or W + 11 = 0

W = 7 or W = - 11

Since the width cannot be negative, then W = 7cm

L = 77/7 = 11 cm

4 0
3 years ago
Other questions:
  • Sean has some candy bars that he wants to give away. he is going to give each person 1/8 of a bar, and he has 2 3/4 bars to give
    5·2 answers
  • What is the interquartile range of the data set? 23, 35, 55, 61, 64, 67, 68, 71, 75, 94, 99
    12·2 answers
  • Adrianna can read 7 pages in 10 minutes at this rate how many can she read in 25 minutes
    14·2 answers
  • What is the total cost for a $20 haircut with a 15% tip
    12·2 answers
  • How do you solve (Y-2)(y+4)
    12·1 answer
  • Plz can someone help
    14·1 answer
  • Hi please help me with this
    15·1 answer
  • the students hold a carwash. They charge $5 per car. if they wash 23 card and use the funds to repay the school what remains of
    10·1 answer
  • HELPPPP!!!! geometry plzzz..... <br> What is the distance between (3,-4) and (8,8) ?
    6·1 answer
  • Find the range from the following date 50,75,65,35,85,55,95,70,25,45​
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!