Calculus HW

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Miran Anwar, mth 235 ss21 1: Hw12-3.4-LT-GS. Due: 04/26/2021 at 11:00pm EDT.

See in LN, § 3.4, See Theorem 3.4.6. 1. (10 points) Use the definition of the Laplace Transform and the properties of the Dirac Delta generalized function δ to compute the following expressions.

(a) L [ −8δ(t −4)

] =

(b) L [ −3δ(t + 2)

] =

(d) L [ 3 cos(t)δ(t −π)

] =

See in LN, § 3.4, See Examples 3.4.4-3.4.6. 2. (10 points)

Consider the initial value problem for function y given by,

y′′−4 y′+ 3 y =−δ(t −1), y(0) = 0, y′(0) = 0.

(a) Find the Laplace Transform of the source function, F(s) = L [ −δ(t −1)

] .

F(s) =

(b) Find the Laplace Transform of the solution, Y (s) = L [ y(t)

] .

Y (s) =

y(t) =

Recall: If needed, the step function at c is denoted as u(t −c).

See in LN, § 3.4, See Examples 3.4.4-3.4.6. 3. (10 points)

Consider the initial value problem for function y given by,

y′′+ y′−6 y = 5 t 2 δ(t −2), y(0) = 0, y′(0) = 0.

(a) Find the Laplace Transform of the source function, F(s) = L [ 5 t 2 δ(t −2)

] .

F(s) = 1

(b) Find the Laplace Transform of the solution, Y (s) = L [ y(t)

] .

Y (s) =

y(t) =

Recall: If needed, the step function at c is denoted as u(t −c).

See in LN, § 3.4, See Examples 3.4.4-3.4.6. 4. (10 points)

Consider the initial value problem for function y given by,

y′′−4 y′+ 5 y = 4 δ(t −1), y(0) = 0, y′(0) = 0.

(a) Find the Laplace Transform of the source function, F(s) = L [ 4 δ(t −1)

] .

F(s) =

(b) Find the Laplace Transform of the solution, Y (s) = L [ y(t)

] .

Y (s) =

y(t) =

Recall: If needed, the step function at c is denoted as u(t −c).

See in LN, § 2.1. 5. (10 points)

Note: You have only 3 attempts to do this problem.

Consider a mass-spring system that oscillates with frequency ω when there are no external forces acting on it. Now, suppose that the mass attached to the spring is initially 2 cm and then the mass is let go with an initial velocity of 9 cm/s. Furthermore, at time t = 2 s the mass is struck, once, with a hammer force of 9 grs cm/sˆ2.

The situation above can be described by the initial value problem

y′′ + ω2 y = K δ(t −c), y(0) = y0, y′(0) = y1, 2

where, as usual, y(t) is the displacememnt of the spring from the equilibrium position in cm, positive down- wards, at the time t in seconds, and the hammer strike is modeled by a Dirac’s delta generalized function.

Find the parameters K, c, y0 and y1.

K = c =

y0 = y1 =

Note: You do not need to solve the differential equation. You only need to find the force and write the initial conditions.

See in LN, § 2.1. 6. (10 points)

Note: You have only 5 attempts in this problem. Find the solutions to all parts and then click on “Check Answers”. In this way you get feedback for all parts and you do not waste attempts.

Motivation: The idea behind this problem is figure out how we can use a hammer to stop an oscillating mass-spring system. We need to hit the oscillating mass with the hammer at the right time and with the right intensity so that the mass-spring system stops.

Part 1: The Mass-Spring System Find the solution y(t) of the initial value below, which describies the movement of a mass-spring system that is hit by an impulsive force at the time t = π2 ,

y′′+ y = F0 δ (

t − π

) , y(0) = y0, y

′(0) = 0,

where y0 and F0 are arbitrary constants.

Note: Recall the identity sin(θ− π2 ) =−cos(θ).

• A. y(t) = F0 cos(t)+ y0 u (

t − 3π2 )

cos(t)

• B. y(t) = y0 cos(t)+ F0 u (

t − π2 )

cos(t)

• C. y(t) = y0 sin(t)−F0 u (

t − π2 )

sin(t)

• D. y(t) = y0 sin(t)+ F0 u (

t − π2 )

sin(t)

• E. y(t) = F0 cos(t)−y0 u (

t − π2 )

cos(t)

• F. y(t) = F0 sin(t)−y0 u (

t − π2 )

cos(t)

• G. y(t) = y0 cos(t)−F0 u (

t − π2 )

cos(t)

• H. y(t) = F0 sin(t)+ y0 u (

t − 3π2 )

sin(t) • I. None of the above.

Part 2: Stoping the Spring with an Impulsive Force See in LN, § 3.4, See Examples 3.4.4-3.4.6. 7. (0 points)

Consider the initial value problem for function y given by,

y′′−4 y′+ 4 y = δ(t −1), y(0) = 0, y′(0) = 0.

(a) Find the Laplace Transform of the source function, F(s) = L [

δ(t −1) ] .

F(s) =

(b) Find the Laplace Transform of the solution, Y (s) = L [ y(t)

] .

Y (s) =

y(t) =

Recall: If needed, the step function at c is denoted as u(t −c).

See in LN, § 3.4, See Examples 3.4.4-3.4.6. 8. (0 points)

Consider the initial value problem for function y given by,

y′′+ 2 y′+ 5 y = 2 δ(t −3), y(0) = 0, y′(0) = 0.

(a) Find the Laplace Transform of the source function, F(s) = L [ 2 δ(t −3)

] .

F(s) =

(b) Find the Laplace Transform of the solution, Y (s) = L [ y(t)

] .

Y (s) =

y(t) =

Recall: If needed, the step function at c is denoted as u(t −c).