How many integers from 1 to 1 million dont have common factors with 2020

) Use the Laplace transform to solve the following initial value problem: y′′−6y′+9y=0y(0)=4,y′(0)=2 Using Y for the Laplace tra

artcher [175]

Answer:

$y(t)=2e^{3t}(2-5t)$

Step-by-step explanation:

Let Y(s) be the Laplace transform Y=L{y(t)} of y(t)

Applying the Laplace transform to both sides of the differential equation and using the linearity of the transform, we get

L{y'' - 6y' + 9y} = L{0} = 0

(*) L{y''} - 6L{y'} + 9L{y} = 0 ; y(0)=4, y′(0)=2

Using the theorem of the Laplace transform for derivatives, we know that:

$\large\bf L\left\{y''\right\}=s^2Y(s)-sy(0)-y'(0)\\\\L\left\{y'\right\}=sY(s)-y(0)$

Replacing the initial values y(0)=4, y′(0)=2 we obtain

$\large\bf L\left\{y''\right\}=s^2Y(s)-4s-2\\\\L\left\{y'\right\}=sY(s)-4$

and our differential equation (*) gets transformed in the algebraic equation

$\large\bf s^2Y(s)-4s-2-6(sY(s)-4)+9Y(s)=0$

Solving for Y(s) we get

$\large\bf s^2Y(s)-4s-2-6(sY(s)-4)+9Y(s)=0\Rightarrow (s^2-6s+9)Y(s)-4s+22=0\Rightarrow\\\\\Rightarrow Y(s)=\frac{4s-22}{s^2-6s+9}$

Now, we brake down the rational expression of Y(s) into partial fractions

$\large\bf \frac{4s-22}{s^2-6s+9}=\frac{4s-22}{(s-3)^2}=\frac{A}{s-3}+\frac{B}{(s-3)^2}$

The numerator of the addition at the right must be equal to 4s-22, so

A(s - 3) + B = 4s - 22

As - 3A + B = 4s - 22

we deduct from here

A = 4 and -3A + B = -22, so

A = 4 and B = -22 + 12 = -10

It means that

$\large\bf \frac{4s-22}{s^2-6s+9}=\frac{4}{s-3}-\frac{10}{(s-3)^2}$

and

$\large\bf Y(s)=\frac{4}{s-3}-\frac{10}{(s-3)^2}$

By taking the inverse Laplace transform on both sides and using the linearity of the inverse:

$\large\bf y(t)=L^{-1}\left\{Y(s)\right\}=4L^{-1}\left\{\frac{1}{s-3}\right\}-10L^{-1}\left\{\frac{1}{(s-3)^2}\right\}$

we know that

$\large\bf L^{-1}\left\{\frac{1}{s-3}\right\}=e^{3t}$

and for the first translation property of the inverse Laplace transform

$\large\bf L^{-1}\left\{\frac{1}{(s-3)^2}\right\}=e^{3t}L^{-1}\left\{\frac{1}{s^2}\right\}=e^{3t}t=te^{3t}$

and the solution of our differential equation is

$\large\bf y(t)=L^{-1}\left\{Y(s)\right\}=4L^{-1}\left\{\frac{1}{s-3}\right\}-10L^{-1}\left\{\frac{1}{(s-3)^2}\right\}=\\\\4e^{3t}-10te^{3t}=2e^{3t}(2-5t)\\\\\boxed{y(t)=2e^{3t}(2-5t)}$

5 0

3 years ago

Question 1 of 10<br> (x)=x2. What is g(x)?<br> y<br> f(x)<br> g(x)<br> -5<br> 6<br> (3, 3)<br> 5

never [62]

The graph is attached and the answer to the function g(x) = (1/3)x²

Option A is the right answer.

<h3>What is a Function ?</h3>

A function is a mathematical statement that relates a dependent variable and an independent variable.

It is given that

f(x) = x²

It has been asked to determine the equation of g(x)

It can be seen from the graph the the vertex is same just the graph is scaled.

The point (3,3) is of g(x) , when the value of y at at x =3 is 9

Therefore the scale factor is 1/3

And we can do trial and error on the basis of the options given ,

The graph is attached and the answer to the function g(x) = (1/3)x²

Option A is the right answer.

To know more about Function

brainly.com/question/12431044

#SPJ1

6 0

1 year ago

Now Cathy is three times as old as Katie. In six years Cathy will be only twice as old as Katie. How old is each girl now?

BabaBlast [244]

Katie is 6; Cathy is 18!

X- Katie’s age now and 3x -Cathys age now
In 6 years, Katie is 12 and Cathy is 24.

2 x (X+6) = 3x + 6
2x + 12 = 3x + 6
6 = x

6 0

3 years ago

I NEED HELP QUICK ASAP!

kramer

Triangle 1 area : 6×4 /2 -> 24/2 -> 12
triangle 2 area : 9×8 /2 -> 72/2 -> 36
rectangle 1 area : 6×4 -> 24
rectangle 2 area : 15×4 -> 60
12+36+24+60
48+24+60
72+60
132

8 0

3 years ago

Read 2 more answers

HELPPPPPP i'm taking a test right now

dalvyx [7]

Answer:

AE

Step-by-step explanation:

pls make me brainliest

3 0

2 years ago