What does -$20 -$15 +$30 $2 Equal

(10 points) Consider the initial value problem y′+3y=9t,y(0)=7. Take the Laplace transform of both sides of the given differenti

Rashid [163]

Answer:

The solution

$Y (s) = 9( -1 +3 t + e^{-3 t} ) + 7 e ^{-3 t}$

Step-by-step explanation:

Explanation:-

Consider the initial value problem y′+3 y=9 t,y(0)=7

Step(i):-

Given differential problem

y′+3 y=9 t

Take the Laplace transform of both sides of the differential equation

L( y′+3 y) = L(9 t)

Using Formula Transform of derivatives

L(y¹(t)) = s y⁻(s)-y(0)

By using Laplace transform formula

$L(t) = \frac{1}{S^{2} }$

Step(ii):-

Given

L( y′(t)) + 3 L (y(t)) = 9 L( t)

$s y^{-} (s) - y(0) + 3y^{-}(s) = \frac{9}{s^{2} }$

$s y^{-} (s) - 7 + 3y^{-}(s) = \frac{9}{s^{2} }$

Taking common y⁻(s) and simplification, we get

$( s + 3)y^{-}(s) = \frac{9}{s^{2} }+7$

$y^{-}(s) = \frac{9}{s^{2} (s+3}+\frac{7}{s+3}$

Step(iii):-

By using partial fractions , we get

$\frac{9}{s^{2} (s+3} = \frac{A}{s} + \frac{B}{s^{2} } + \frac{C}{s+3}$

$\frac{9}{s^{2} (s+3} = \frac{As(s+3)+B(s+3)+Cs^{2} }{s^{2} (s+3)}$

On simplification we get

9 = A s(s+3) +B(s+3) +C(s²) ...(i)

Put s =0 in equation(i)

9 = B(0+3)

B = 9/3 = 3

Put s = -3 in equation(i)

9 = C(-3)²

C = 1

Given Equation 9 = A s(s+3) +B(s+3) +C(s²) ...(i)

Comparing 'S²' coefficient on both sides, we get

9 = A s²+3 A s +B(s)+3 B +C(s²)

0 = A + C

put C=1 , becomes A = -1

$\frac{9}{s^{2} (s+3} = \frac{-1}{s} + \frac{3}{s^{2} } + \frac{1}{s+3}$

Step(iv):-

$y^{-}(s) = \frac{9}{s^{2} (s+3}+\frac{7}{s+3}$

$y^{-}(s) =9( \frac{-1}{s} + \frac{3}{s^{2} } + \frac{1}{s+3}) + \frac{7}{s+3}$

Applying inverse Laplace transform on both sides

$L^{-1} (y^{-}(s) ) =L^{-1} (9( \frac{-1}{s}) + L^{-1} (\frac{3}{s^{2} }) + L^{-1} (\frac{1}{s+3}) )+ L^{-1} (\frac{7}{s+3})$

By using inverse Laplace transform

$L^{-1} (\frac{1}{s} ) =1$

$L^{-1} (\frac{1}{s^{2} } ) = \frac{t}{1!}$

$L^{-1} (\frac{1}{s+a} ) =e^{-at}$

Final answer:-

Now the solution , we get

$Y (s) = 9( -1 +3 t + e^{-3 t} ) + 7 e ^{-3t}$

5 0

3 years ago

Factor. z^2 -6z -16 simplify

leva [86]

(z-8) (z+2)
What I do to factor is find _x_=-16 and _+_=-6. -8x2=-16 and -8+2=-16. The coordinates for z if you were to graph this would be (8,0) and (-2,0)

3 0

3 years ago

8.0464x10^-7 = answer in decimal notation

Charra [1.4K]

To convert 8.0464x10^-7 from scientific notation to decimal notation, you need to move the decimal point to the left seven times.
Therefore the answer is:

0.00000080464

Note: 10^-7 equals 1/10^7
8.0464x10^-7 can be rewritten as:
8.0464/10^7 (i.e. divide 8.0464 by 10000000)
which results: 0.00000080464

6 0

3 years ago

If a and b are two angles in standard position in Quadrant I, find cos(a+b) for the given function values. sin a=4/5 and cos b=5

vladimir1956 [14]

First let's find the angles a and b.

We have then:
sin a = 4/5
a = Asin (4/5)
a = 53.13 degrees.

cos b = 5/13
b = Acos5 / 13
b = 67.38 degrees.

We now calculate cos (a + b). To do this, we replace the previously found values:
cos ((53.13) + (67.38)) = - 0.507688738
Answer:
-0.507688738
Note: there is another way to solve the problem using trigonometric identities.

4 0

3 years ago

Read 2 more answers

If a linear function contains the pairs (5,7) and (9,11), which pairs should have the reciprocal function to this one?

OLga [1]

The new coordinates will be ( 7,5) and (11,9) , Option A is the correct answer.

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

The reciprocal function is the inverse of the function, such that 1/ f(x) is the reciprocal of f(x).

The given linear function has pairs of points

(5,7) and (9,11)

The reciprocal of this will be

interchanging of the coordinates

Therefore the new coordinates will be

( 7,5) and (11,9)

Option A is the correct answer.

To know more about Reciprocal Function

brainly.com/question/12621604

#SPJ1

8 0

2 years ago