use the Laplace transform to solve the given initial-value problem.y"+y=f(t)y(0)=0 , y'(0)=1 wheredisplaystyle f{{left({t}right)}}={leftlbracebegin{matrix}{1}&{0}le{t}<frac{pi}{{2}}{0}&{t}gefrac{pi}{{2}}end{matrix}right.}

Question

use the Laplace transform to solve the given initial-value problem.y"+y=f(t)y(0)=0 , y'(0)=1 wheredisplaystyle f{{left({t}right)}}={leftlbracebegin{matrix}{1}&{0}le{t}<frac{pi}{{2}}{0}&{t}gefrac{pi}{{2}}end{matrix}right.}

bobbie71G 2021-03-04 Answered

use the Laplace transform to solve the given initial-value problem.
$y " + y = f (t)$
$y (0) = 0, y^{'} (0) = 1$ where
$f (t) = {\begin{matrix} 1 & 0 \leq t < \frac{π}{2} \\ 0 & t \geq \frac{π}{2} \end{matrix}$

Ask Expert 1 See Answers

You can still ask an expert for help

Expert Answer

SchulzD

Answered 2021-03-05 Author has 83 answers

Step 1
Given, the initial value problem
$y " + y = f (t)$
$y (0) = 0, y^{'} (0) = 1$ where
$f (t) = {\begin{matrix} 1 & 0 \leq t < \frac{π}{2} \\ 0 & t \geq \frac{π}{2} \end{matrix}$
step 2
The Laplace transform of the two derivatives
$L {y^{'}} = s Y (s) - y (0)$
$L {y} = s^{2} Y (s) - s y (0) - y^{'} (0)$
$L {1} = \frac{1}{s}$
$L {e^{a t}} = \frac{1}{s - a}$
Where y(0) and y’(0) are initial conditions.
Step 3
Now,
when $0 \leq t \leq \frac{π}{2}$
$y " + y^{'} = 1, y (0) = 0 . y^{'} (0) = 1$
$s^{2} Y (s) - s y (0) - y^{'} (0) + s Y (s) - y (0) = 1$
${s^{2} + s} Y (s) - s \cdot 0 - 1 - 0 = 1$
${s^{2} + s} Y (s) = 2$
$Y (s) = \frac{2}{s^{2} + s} = \frac{2}{s (s + 1)} = 2 {\frac{1}{s} - \frac{1}{s + 1}}$
$Y (s) = 2 {1 - e^{- t}}$
when $t \geq \frac{π}{2}$
$y " + y^{'} = 0, y (0) = 0, y^{'} (0) = 1$
$s^{2} Y (s) - s y (0) - y^{'} (0) + s Y (s) - y (0) = 1$
${s^{2} + s} Y (s) - s \cdot 0 - 1 - 0 = 0$
${s^{2} + s} Y (s) = 1$
$Y (s) = \frac{1}{s^{2} + s} = \frac{1}{s (s + 1)} = {\frac{1}{s} - \frac{1}{s + 1}}$
$Y (s) = {1 - e^{- t}}$
Step 4
Hence the solution is

This is helpful 68

Relevant Questions

asked 2021-06-01

Find the vectors T, N, and B at the given point.

r (t) =< t^{2}, \frac{2}{3} t^{3}, t >

and point

< 4, - \frac{16}{3}, - 2 >

See answers (2)

asked 2021-05-14

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR
Consider the points below
P(1,0,1) , Q(-2,1,4) , R(7,2,7)
a) Find a nonzero vector orthogonal to the plane through the points P,Q and R
b) Find the area of the triangle PQR

See answers (2)

asked 2021-09-22

Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at $(1, 3, 0), (- 2, 0, 2), and (- 1, 3, - 1)$ .

See answers (1)

asked 2022-03-09

vv

A second hand car dealer has ten cars for sale. He decides to investigate the link between the

ages of the cars, x years and the mileage, y thousand miles. The data collected from the cars

are shown in the table below.

Age, x years

2 2.5 3 4 4.5 4.5 5 3 6 6.5

Mileage, y thousand miles

22 34 33 37 40 45 49 30 58 58

(a) Construct a well labeled scatter diagram in the graph paper provided to present the

information. (3 marks)

(b) Find Sxx and Sxy. (3 marks)

(c) Find the regression equation in the form y = α + βx. (4 marks)

(d) Fit the regression equation obtained in (c) to the scatter plot in (a) and label the line

appropriately. (2 marks)

(e) Give a practical interpretation of the slope β. (1 mark)

(f) Using your answer in (c), fifind the mileage for a 6 year old car.

A second hand car dealer has ten cars for sale. He decides to investigate the link between the

ages of the cars, x years and the mileage, y thousand miles. The data collected from the cars

are shown in the table below.

Age, x years

2 2.5 3 4 4.5 4.5 5 3 6 6.5

Mileage, y thousand miles

22 34 33 37 40 45 49 30 58 58

(a) Construct a well labeled scatter diagram in the graph paper provided to present the

information. (3 marks)

(b) Find Sxx and Sxy. (3 marks)

(c) Find the regression equation in the form y = α + βx. (4 marks)

(d) Fit the regression equation obtained in (c) to the scatter plot in (a) and label the line

appropriately. (2 marks)

(e) Give a practical interpretation of the slope β. (1 mark)

(f) Using your answer in (c), fifind the mileage for a 6 year old car.

A second hand car dealer has ten cars for sale. He decides to investigate the link between the

ages of the cars, x years and the mileage, y thousand miles. The data collected from the cars

are shown in the table below.

Age, x years

2 2.5 3 4 4.5 4.5 5 3 6 6.5

Mileage, y thousand miles

22 34 33 37 40 45 49 30 58 58

(a) Construct a well labeled scatter diagram in the graph paper provided to present the

information. (3 marks)

(b) Find Sxx and Sxy. (3 marks)

(c) Find the regression equation in the form y = α + βx. (4 marks)

(d) Fit the regression equation obtained in (c) to the scatter plot in (a) and label the line

appropriately. (2 marks)

(e) Give a practical interpretation of the slope β. (1 mark)

(f) Using your answer in (c), fifind the mileage for a 6 year old car

See answers (1)

asked 2021-09-15

Find the principal unit normal vector to the curve at the specified value of the parameter

r (t) = t i + \frac{6}{t} j, t = 3

See answers (1)

asked 2021-03-12

Let $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ and 'k' be the scalar. Find the formula that relates 'detKA' to 'K' and 'detA''

See answers (1)

asked 2021-12-03

Find the distance from the origin to the line
x=1+t, y=2-t, z=-1+2t

See answers (1)

Algebra

Linear algebra

use the Laplace transform to solve the given initial-value problem.y"+y=f(t)y(0)=0 , y'(0)=1 wheredisplaystyle f{{left({t}right)}}={leftlbracebegin{matrix}{1}&{0}le{t}<frac{pi}{{2}}{0}&{t}gefrac{pi}{{2}}end{matrix}right.}

Expert Answer

Not exactly what you’re looking for?

Relevant Questions