Laplace Tutorial PDF

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TutorialExercises Laplace Transforms (taken from textbook – Exercise 11.2.6, Q1 – Q3)

Q1.

Using the definition of the Laplace Transform (i.e. do the integration) : 

ℒ() =  ()    

obtain the transforms of the functions f(t) when f(t) is given by : (a) (b) (c) (d) Q2.

2t -2t [n.b. cosh(2t) = (e + e ) / 2 ]

What are the abscissa of convergence for the following functions? (i.e. s > ?) (b) (c) (d) (f)

Q3.

cosh 2t 2 t 3+t -t te

e -3t sin 2t sinh 3t t4

(g) e- t +t2 (h) 3 cos 2t – t3 (j) sinh 3t + sin 3t

Using the results from the table of Laplace Transforms, obtain the LT’s of the following functions and state the region of convergence : (a) (c) (d) (e) (f)

5 – 3t 3 – 2t + 4 cos 2t cosh 3t sinh 2t 5e-2t + 3 – 2 cos 2t

(g) 4 te-2t (i) t2 e-4t (j) 6t 3– 3t 2 + 4t – 2 (k) 2 cos 3t + 5 sin 3t

Inverse Laplace Transforms (taken from textbook – Exercise 11.2.10, Q4)

Q4.

Find the inverse Laplace Transform L-1{F(s)} of the function F(s) when F(s) is given by : (b)

s 5 s ( 1)(s 3)

(j)

3 s2 7 s 5 (s 1)(s 2)(s 3)

(c)

s 1 s (s 3)

(k)

5s 7 (s 3)(s2 2)

(d)

2s 6 s2 4

(l)

s ( s 1)( s2 2 s 2)

(f)

s 8 s 4s 5

(n)

s 1 (s 2)( s 3)( s 4)

(h)

4s ( s 1)( s 1)2

(o)

3s ( s 1)( s2 4)

2

2

LT Methods for Solving ODE’s (taken from MEM textbook – Exercise 11.3.4, Q5) Q5

Solve the following differential equations for t the stated initial conditions. (a)

dx 3x dt

(c)

d2x dx 2 5 x 1 2 dt dt

(e)

d2x dx 3 2 dt dt

(g)

(h)

e

d2x dt 2

dx dt

d2y dt 2

2

when t = 0, x=2

2t

2x

2x

0 using the Laplace transform method together with

when t = 0, x = 0, dx/dt = 0

2e

t

4t

5e sin(t )

dy 3 y 3t dt

when t = 0, x = 0, dx/dt = 1

when t = 0, x = 1, dx/dt = 0

when t = 0, y = 0, dy/dt = 1...