Consider the differential equation:


2y'' + ty' − 2y = 14, y(0) = y'(0) = 0.


In some instances, the Laplace transform can be used to solve linear differential equations with variable monomial coefficients.


If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . ,then

ℒ{tnf(t)} = (-1)^n d^n/ds^n F(s)


to reduce the given differential equation to a linear first-order DE in the transformed function Y(s) = ℒ{y(t)}.


Requried:

a. Sovle the first order DE for Y(s).

b. Find find y(t)= ℒ^-1 {Y(s)}

Answers

Answer 1

(a) Take the Laplace transform of both sides:

[tex]2y''(t)+ty'(t)-2y(t)=14[/tex]

[tex]\implies 2(s^2Y(s)-sy(0)-y'(0))-(Y(s)+sY'(s))-2Y(s)=\dfrac{14}s[/tex]

where the transform of [tex]ty'(t)[/tex] comes from

[tex]L[ty'(t)]=-(L[y'(t)])'=-(sY(s)-y(0))'=-Y(s)-sY'(s)[/tex]

This yields the linear ODE,

[tex]-sY'(s)+(2s^2-3)Y(s)=\dfrac{14}s[/tex]

Divides both sides by [tex]-s[/tex]:

[tex]Y'(s)+\dfrac{3-2s^2}sY(s)=-\dfrac{14}{s^2}[/tex]

Find the integrating factor:

[tex]\displaystyle\int\frac{3-2s^2}s\,\mathrm ds=3\ln|s|-s^2+C[/tex]

Multiply both sides of the ODE by [tex]e^{3\ln|s|-s^2}=s^3e^{-s^2}[/tex]:

[tex]s^3e^{-s^2}Y'(s)+(3s^2-2s^4)e^{-s^2}Y(s)=-14se^{-s^2}[/tex]

The left side condenses into the derivative of a product:

[tex]\left(s^3e^{-s^2}Y(s)\right)'=-14se^{-s^2}[/tex]

Integrate both sides and solve for [tex]Y(s)[/tex]:

[tex]s^3e^{-s^2}Y(s)=7e^{-s^2}+C[/tex]

[tex]Y(s)=\dfrac{7+Ce^{s^2}}{s^3}[/tex]

(b) Taking the inverse transform of both sides gives

[tex]y(t)=\dfrac{7t^2}2+C\,L^{-1}\left[\dfrac{e^{s^2}}{s^3}\right][/tex]

I don't know whether the remaining inverse transform can be resolved, but using the principle of superposition, we know that [tex]\frac{7t^2}2[/tex] is one solution to the original ODE.

[tex]y(t)=\dfrac{7t^2}2\implies y'(t)=7t\implies y''(t)=7[/tex]

Substitute these into the ODE to see everything checks out:

[tex]2\cdot7+t\cdot7t-2\cdot\dfrac{7t^2}2=14[/tex]


Related Questions

Find n for the arithmetic sequence for which sn=345, u1=12 and d = 5 .

Answers

Answer:

n = 10

Step-by-step explanation:

The sum to n terms of an arithmetic sequence is

[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [ 2a₁ + (n - 1)d ]

where a₁ is the first term and d the common difference

Here a₁ = 12 and d = 5 and [tex]S_{n}[/tex] = 345, thus

[tex]\frac{n}{2}[/tex] [ (2 × 12) + 5(n - 1) ] = 345 ( multiply both sides by 2 )

n( 24 + 5n - 5) = 690 ← distribute and simplify left side

n(19 + 5n) = 690

19n + 5n² = 690 ( subtract 690 from both sides )

5n² + 19n - 690 = 0 ← in standard form

(5n + 69)(n - 10) = 0 ← in factored form

Equate each factor to zero and solve for n

5n + 69 = 0 ⇒ 5n = - 69 ⇒ n = - [tex]\frac{69}{5}[/tex]

n - 10 = 0 ⇒ n = 10

However, n > 0 , thus n = 10

The local bowling alley pays you
$7.25 per hour to manage the desk.
Last week you worked 16 hours.
What is your straight-time pay?

Answers

Answer:

my straight time payment will be $116 for last week

Step-by-step explanation:

The local bowling alley pays

$7.25 per hour to manage the desk.

If worked 16 hours, my straight time payment will be

Rate= $7.25 per hour

Hour worked= 16 hours

my straight time payment = rate*hour worked

my straight time payment = 7.25*16

my straight time payment = 166.00

my straight time payment will be $116

3.85∙47.3+52.7∙3.85 PLSSSSS HELP

Answers

Answer:

385

Step-by-step explanation:

What is the area of the house (including the drawing room, TV room, balcony, hallway, kitchen, and bedroom)?

Answers

Answer:

1256 i think

Step-by-step explanation:

in the diagram,QOS and ROU are straight lines.OT is the bisector of angle UOS. Angle POQ and angle QOR are complementary angles. Find the values of x and y.​pleaseeee answer sooonnn

Answers

Answer:

x=50° and y=45°

Step-by-step explanation:

x=QU(90°)-QP(40°)

x=50°

y=SU(90°)/2

y=45°

PLEASE ANSWER!!! Brainliest to whoever answers at first!Order the following expressions by their values from least to greatest.

Answers

Answer:

  -j, 0, j-k

Step-by-step explanation:

j is a positive number, so -j will be less than 0.

j is a number greater than k, so j - k will be greater than 0.

From least to greatest, the order is ...

  -j, 0, j-k

Answer:

Step-by-step explanation:

please help solving.​

Answers

Answer:

right machine first, then left.6 into left machine, then right

Step-by-step explanation:

a) Putting 6 into the first (left) machine will give an output of ...

  y = √(6 -5) = √1 = 1

Putting 1 into the second (right) machine will give an output of ...

  y = 1² -6 = -5

This answers the second question, but not the first question.

__

If we put 6 into the right machine first, we get an output of ...

  y = 6² -6 = 30

Putting 30 into the left machine, we get an output of ...

  y = √(30 -5) = √25 = 5 . . . . . the desired output.

The input must go into the right machine first, then its output goes into the left machine to get a final output of 5 from an input of 6.

__

b) The left machine cannot produce negative outputs, so achieving an output of -5 with the arrangement used in part A is not possible. (green curves in the attached graph)

However, as we have shown above, inputting 6 to the left machine first, following that by processing with the right machine, can produce an output of -5. (purple curve in the attached graph)

Examine the diagram. Triangle M P L. Angle P is 90 degrees and angle L is (4 x + 6) degrees. The exterior angle to angle M is 136 degrees. What is the value of x?

Answers

Answer:

x = [tex]10^{0}[/tex]

Step-by-step explanation:

From Δ MPL given that; <P = [tex]90^{0}[/tex], exterior angle to M = [tex]136^{0}[/tex] and <L = [tex](4x+6)^{0}[/tex].

In the triangle, the exterior angle is equal to the sum of the two adjacent interior angles. So that;

[tex]136^{0}[/tex] = [tex]90^{0}[/tex] + [tex](4x+6)^{0}[/tex]

      = [tex]90^{0}[/tex] + 4[tex]x^{0}[/tex] + [tex]6^{0}[/tex]

[tex]136^{0}[/tex] = [tex]96^{0}[/tex] + [tex]x^{0}[/tex]

⇒      4[tex]x^{0}[/tex] = [tex]136^{0}[/tex] - [tex]96^{0}[/tex]

        4[tex]x^{0}[/tex]  = [tex]40^{0}[/tex]

Divided both sides by 4 to have;

[tex]x^{0}[/tex] = [tex]10^{0}[/tex]

The value of x is [tex]10^{0}[/tex].

Answer:

The correct answer is in fact 10 degrees.

Step-by-step explanation:

I hope this helps!

Have a great day! :)

Suppose x varies directly with the square root of y and inversely with the cube root of z. What equation models this combined variation?

Answers

Answer:

[tex]\huge\boxed{x = k \frac{\sqrt{y} }{\sqrt[3]{z} }}[/tex]

Step-by-step explanation:

Given that:

1) x ∝ √y

2) x ∝ [tex]\frac{1}{\sqrt[3]{z} }[/tex]

Combining the proportionality

=> x ∝ [tex]\frac{\sqrt{y} }{\sqrt[3]{z} }[/tex]

=> [tex]x = k \frac{\sqrt{y} }{\sqrt[3]{z} }[/tex]

Where k is the constant of proportionality.

To which set of numbers does the number sqr rt-16 belong? Select all that apply

Answers

Answer:

The square root of -16 is an imaginary number and a complex number. Sqrt(-16)=4i. We use the i to indicate that the number is imaginary since there is no number that can be multiplied by itself to get a negative number (a negative times a negative is a positive, and a positive times a positive is also a positive). So the use of i tells you immediately that it's an imaginary number. You can tell the number is complex because it has both a real and an imaginary part and could be written in the form a+bi, where a is a real number and bi is an imaginary number. In this specific case, the real part (a) is 0 and the imaginary part (bi) is 4i.

Step-by-step explanation:

Find the sum of (5x3 + 3x2 - 5x + 4) and (8x3 -5x2 + 8x + 9)

Answers

Answer:

(5x³+3x²-5x+4) + (8x³-5x²+8x+9)

= 5x³+3x²-5x+4 +8x³-5x²+8x+9

= 5x³+8x³+3x²-5x²-5x+8x+4+9

= 13x³-2x²+3x+13

Hope this helps

if u have question let me know in comments ^_^

ASAP HELP WILL MARK BRAINLIEST

Answers

Answer:

c(x)=(3/4)^x

(3/4)^-2= 16/9

(3/4)^-1 =4/3

(3/4)^0=1

(3/4)^1 = 3/4

(3/4)^2= 9/16

Musah stands at the centre of a rectangular field. He first takes 50 steps north, then 25 steps
west and finally 50 steps on a bearing of 3150
.
i. Sketch Musah’s movement [Mark 4]
ii. How far west is Musah’s final point from the centre?

Answers

Answer:

Inokkohgy8uokokj76899

(Score for Question 1: ____ of 5 points)
1. The equation of a circle is x2 + y2 - 4x + 2y - 11 = 0. What are the center and the radius of the circle?
Show your work
Answer:)

Answers

Answer:

The center of the circle:

(2, -1)

The radius of the circle:

r = 4

Step-by-step explanation:

The first step is to complete the square on both x and y:

x^2 + y^2 -4x + 2y - 11 = 0

x^2 -4x +y^2 +2y = 11

We determine c1 to complete the square on x:

c1 = (b/2)^2

c1 = (-4/2)^2 = 4

We then determine c2 to complete the square on y:

c2 = (2/2)^2 = 1

The equation of the circle is then re-written as:

x^2 -4x + 4 +y^2 +2y + 1 = 11 +4 +1

We then factorize the expressions in x and y separately:

(x-2)^2 + (y +1)^2 = 16

The center of the circle is thus:

(2, -1)

The radius is the square root of 16:

r = 4

Answer:

center :  (2, -1)

radius :  r = 4

Step-by-step explanation:

Which quadratic equation would be used to solve for the unknown dimensions?

0 = 2w2
512 = w2
512 = 2w2
512 = 2l + 2w

Answers

Answer:

C

Step-by-step explanation:

Answer:

C: 512 = 2w2

Step-by-step explanation:

on edge

36x7 please EXPLAIN the process of the multiplication plse

Answers

36×7

=252

Explaination :

First Multiply 6 and 7 we get 42 !

Write 2 and 4 will be added to the product of 3×7

We get 21 and add 4 here

So we get 252

Answer:

[tex]36 \times 7 = 252[/tex]

Step-by-step explanation:

Firstly multiply 6 with 7 you have to write 2 and take 4 carry and then multiply 7 with 3 u get 21 now add the number u carry in 21 u get ur answer. 252.

Hope it helps u mate

Are you able to find tri-sector (equally divided by 3) rays for an arbitrary angle with straightedge-and-compass construction?

Answers

Answer:

Step-by-step explanation:

No, it is an ancient problem which has been proved to be impossible (in 1837), at least not for an arbitray angle.

However, we can trisect certain angles, such as 90 degrees, but rather than trisection, we are just constructing 30 degree angles.

For further reading, google "angle trisection"

5(y–3.8)=4.7(y–4) help help

Answers

Answer:

y = 2/3 or 0.667

Step-by-step explanation:

5(y–3.8)=4.7(y–4)

Expand the terms in the bracket

That's

5y - 19 = 4.7y - 18.8

Group like terms

5y - 4.7y = 19 - 18.8

0.3y = 0.2

Divide both sides by 0.3

We have the final answer as

y = 2/3 or 0.667

Hope this helps you

Given f(x) = –2x+5 find f'(x).
a f'(x)=- 5x+1.5
b.
x 5
2 2
f'(x) =
C. f'(x) = 2x-5
d.
x 5
2 2
f'(x) -- +
R
Please select the best answer from the choices provided
B.
ОООО
D

Answers

Answer: D

Step-by-step explanation:

To find the inverse function, you switch y with x and x with y. Then you solve for y.

y=-2x+5               [replace y with x and x with y]

x=-2y+5               [subtract both sides by 5]

x-5=-2y                 [divide both sides by -2]

(x-5)/-2=y

Now that we have our inverse function, we can rewrite it so that it matches the answer choice. D matches our answer choice the best.

4/2÷4/7? plz help me ​

Answers

Answer:

2

Step-by-step explanation:

4/2÷4/7

= 4/2 × 7/4

= 28/14

= 2

Which relation is a function?

Answers

Answer:

The second graph is a function.

Step-by-step explanation:

This is the only one that passes the vertical line test.

(If there exits a vertical line which passes through more than one point, then the relation is NOT a function).

What are the solutions to the system of equations? {y=2x2−8x+5y=x−2 (3.5, 0.5) and (1, −1) (7, 5) and (0.5, −1.5) (3.5, 1.5) and (1, −1) (3.5, 1.5) and (−1, −3)

Answers

Answer:

[tex](1,-1)[/tex] and [tex](3.5,1.5)[/tex]

Step-by-step explanation:

Given

[tex]y = 2x^2 - 8x+5[/tex]

[tex]y = x - 2[/tex]

Required

Determine the solution

Substitute x - 2 for y in [tex]y = 2x^2 - 8x+5[/tex]

[tex]x - 2 = 2x^2 - 8x+5[/tex]

Collect like terms

[tex]0 = 2x^2 - 8x - x + 5 + 2[/tex]

[tex]0 = 2x^2 - 9x + 7[/tex]

Expand the expression

[tex]0 = 2x^2 - 7x - 2x+ 7[/tex]

Factorize

[tex]0 = x(2x - 7) -1(2x - 7)[/tex]

[tex]0 = (x-1)(2x - 7)[/tex]

Split the expression

[tex]x - 1 = 0[/tex] or [tex]2x - 7 = 0[/tex]

Solve for x in both cases

[tex]x = 1[/tex] or [tex]2x = 7[/tex]

[tex]x = 1[/tex] or [tex]2x/2 = 7/2[/tex]

[tex]x = 1[/tex] or [tex]x = 3.5[/tex]

Recall that

[tex]y = x - 2[/tex]

When [tex]x = 1[/tex]

[tex]y = 1 -2[/tex]

[tex]y = -1[/tex]

When [tex]x = 3.5[/tex]

[tex]y = 3.5 - 2[/tex]

[tex]y = 1.5[/tex]

Hence, the solution is;

[tex](1,-1)[/tex] and [tex](3.5,1.5)[/tex]

-50 POINTS- (2/5) please no wrong answers for points. A) y = [tex]\frac{9}{2}[/tex] x + [tex]\frac{1}{2}[/tex] B) y = - [tex]\frac{1}{2} x + \frac{7}{2}[/tex] C) [tex]y = -4x +9[/tex] D) [tex]y=4x+15[/tex]

Answers

This problem is about creating a linear regression model.

First, we should take note of the points:

(-4,8)

(-2,4)

(-1,2)

(1,5)

(2,2)

(6,-5)

(7,6)

It's necessary to find a equation y = ax + b that brings us the least MSE (Mean Squared Error). You can calculate at hand, but I bet it is going to be tiresome.

So, basically intuitively you just need to choose a line that fits closer to the given points.

First: remember if y = ax+b, a is the slope which means if a > 0 the line is " / " and a < 0 the line is " \ ".

A) No, this equation is " / "

B) It could be this one.

C) It could be this one too.

D) Nope. " / "

B) a = -1/2

C) a = -4

You can draw those two lines and see that B) gets closer to the points.

Equation:

Y = -0.4957*X + 3.780

Answer: B)

Answer:

[tex]\Large \boxed{y=-\frac{1}{2} x+\frac{7}{2} }[/tex]

Step-by-step explanation:

Using a graph,

we can see that the line y = -1/2x + 7/2 best fits for the data.

A special mixed-nut blend at a store cost $1.35 per lb, and in 2010 the blend cost $1.83 per lb. Let y represent the cost of a pound of the mixed-nut blend x years after 2005. Use a linear equation model to estimate the cost of a pound of the mixed-nut blend in 2007.

Answers

Answer:

y = $1.542 per lb

Step-by-step explanation:

given data

mixed-nut blend store cost  2005 = $1.35 per lb

blend cost in 2010 = $1.83 per lb

solution

we consider here y = cost of a pound

and x year = after 2005

we will use here linear equation model

so

[tex]\frac{y - 1.35}{1.83-1.35} = \frac{x-10}{5 - 0}[/tex]    .........................1

solve it we get

5y - 6.75 = .48 x

so

at 2007 year here x wil be 2

so

[tex]y = \frac{0.48 \times 2 + 6.75}{5}[/tex]  

solve it we get

y = $1.542 per lb

Musah stands at the center of a rectangular field. He first takes 50 steps north, then 25 steps west and finally 50 steps on a bearing of 315°. Sketch musah's movement. How far west is musah's final point from the center?

Answers

Answer:  4.17 steps

Step-by-step explanation:

Draw a point to use as the center and then sketch 50 units north (up) and 25 units west (left) and 315° which creates a right triangle that has an angle of 360° - 315° = 45°

Use Pythagorean Theorem to find the length of the hypotenuse.

50² + 25² = hypotenuse²   --> hypotenuse = 55.9 units

Since Musah only walked 50 units along the hypotenuse, he is 5.9 units from the center.

Create another right triangle using the remaining 5.9 units as the hypotenuse.  You can use 45°-45°-90° rules OR sin 45° to find the horizontal distance from the center to be 4.17.

see attachment for sketch

Calculate the derivative of the function. Then find the value of the derivative as specified:
f(x) = 5x + 9; f "(2)
A) f "(x) 0,f , (2)-0
B) f , (x)-9; f , (2) = 9
C)f"(x) = 5; f "(2) = 5
D) f '(x) 5x; f '(2) 10

Answers

The correct question is;

Calculate the derivative of the function. Then find the value of the derivative as specified:

f(x) = 5x + 9; f '(2)

A) f'(x) = 0; f'(2) = 0

B) f'(x) = 9; f '(2) = 9

C)f'(x) = 5; f'(2) = 5

D) f '(x) = 5x; f '(2) = 10

Answer:

Option C: f'(x) = 5 and f '(2) = 5

Step-by-step explanation:

We want to find the derivative of f(x) = 5x + 9.

Now, the derivative with respect to x will be;

f'(x) = 5

Now,we also want to find out f'(2)

This means we are to put 2 for x in the derivative function.

In the derivative function, we don't have x as we have just 5.

Thus,f'(2) = 5

convert 407 in base 8 to decimal​

Answers

[tex]4\cdot8^2+0\cdot8^1+7\cdot8^0=256+7=263[/tex]

[tex]407_8=263_{10}[/tex]

What happens to the probability of making a Type II​ error, beta​,as the level of​ significance, alpha​,​decreases? Why?

Answers

Answer:

Lowering the level of significance, α increases the probability of making a Type II error, β.

Step-by-step explanation:

Lowering the level of significance, α increases the probability of making a Type II error, β.

This is because the region of acceptance becomes bigger, and it makes it less likely for one to reject a null hypothesis, when it is false, the type II error.

The dance team is selling headbands to raise
money for dance team jackets. They need
to sell 1,260 headbands. The headbands must
be divided equally among the three coaches.
Each coach is in charge of 10 dancers. If all
the headbands must be sold, how many
headbands will each dancer on the team
need to sell?

Answers

Answer:

42  headbands per dancer

Step-by-step explanation:

Selling 1260 headband

Divide by the three coaches

1260/3

420 per coach

Divide by each dancer under a coach

420/10 = 42

Each dancer must sell 42 headbands

What fraction of a pound is an ounce?

Answers

Answer:

1/16

Step-by-step explanation:

there are 16 ounces in a pound

Answer:

1/16 pounds

Step-by-step explanation:

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