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MATH2012EngineeringMathematics–Workshop11

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Engineering Mathematics Workshop 11: The Laplace Transform II

Overview: We discuss first the Laplace transform method for solving the I.V.P. for linear differential equations with constant coefficients. Then we learn how to write a discontinuous forcing function as a single formula in terms of unit step functions and how to determine its Laplace transform using the operational properties of the Laplace transform in combination with the Basic Table. We also show how to find the original (the discontinuous forcing function) when we are given the Laplace transform of the function using the operational properties of the inverse Laplace transform. Further, we deal with periodic forcing functions such as the square wave function or the sawtooth wave function and consider how to compute Laplace transforms and inverse Laplace transforms of such functions. Finally we introduce the concept of transfer functions as an important input-output representation of linear systems. Learning outcomes You should be able to: 1. Use the Laplace transform to solve the initial value problem for ordinary differential equations with constant coefficients. 2. Represent discontinuous (piece-wise continuous) forcing functions as a single formula in terms of the unit step functions. 3. Compute the Laplace transform and the inverse Laplace transform of discontinuous forcing functions. 4. Compute the Laplace transform and the inverse Laplace transform of periodic forcing functions. 5. Understand the concept of transfer function as an input-output representation of linear time-invariant systems (linear systems with constant coefficients). Exercises 1 Solve the given initial value problems using the method of the Laplace transform (a) y  y  e2t  t; y(0)  1 . (b) (D) y  y  6 y  5e2t ; y (0)  y (0)  0 . (c) y  4 y  4t  8e2t ; y(0)  0 and y (0)  5 . (d) y  7 y  10 y 9 cos t 7 sin t; y(0)  5 and y (0)  4.

2. Sketch the function f (t )  t 2U (t  2) and determine its Laplace transform. 3 Sketch the given function, write it as a single formula in terms of unit step functions and then find its Laplace transform.

0 t 1  0, (a) g( t)    t  2, 1  t

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MATH2012EngineeringMathematics–Workshop11

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0  t 1 1,  (b) (D) f (t )  t 1, 1  t  2 t 2  1, 2  t 

t , 0  t  1 (c) v (t )   0, 1  t  sin t , 0  t  2 (d) g ( t)   2  t  0, 1, 0  t  4 (e) f ( t)    0, 4  t 4 Determine the inverse of the Laplace transform of the given function. (a) (D) F( s) 

e  3s e 2 s  3e 4 s ; (b) G (s )  2 s 9 s 2

5 Solve the given initial value problem using the Laplace transform method. (a) (D) y 2 y 3 y  f( t) with y(0) 0 and y(0)  1, t , 0  t  1 . where the forcing function f (t )is given by f (t )   0, 1  t

(b) x  2 x  2 x U( t 2 )  U( t 4 ); x(0)  1 and x (0)  1. (c) y 4 y  g( t) with y(0) 0 and y(0)  1,where the forcing function is as in Problem 3(d). (d) y   5y   4y  f (t ) with y (0)  0 and y (0)  1, where 0, 0  t  3 f ( t)   2, 3  t 6.

 et , 0  t  1 (a) Sketch the function f (t )   , write it as a single formula in terms  0, 1  t of the unit step function(s) and then determine its Laplace transform. (b) Find the inverse Laplace transform of the expression 1 ( s 1)( s 1)2 2

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MATH2012EngineeringMathematics–Workshop11

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(c) Use the results in (a) and (b) to solve the following initial value problem

y  2 y   y  f ( t) with y(0)  0, y (0)  0, where f (t ) is the function given in (a).  1, 0  t  1 7. Sketch the square wave function f (t )   and f ( t  2)  f (t ) and determine  0, 1 t  2 its Laplace transform. 8 Sketch the saw tooth function f (t )  t , 0  t  2, f (t  2)  f (t )and determine its Laplace transform.  sint , 0  t   9. The half-wave rectification of sine wave is defined as f (t )   and  0,   t  2 f (t  2 )  f (t) . Sketch the function and show that its Laplace transform is L{ f ( t)} 

1 . ( s2  1)(1  e  s )

10 Find the inverse Laplace transform of the given Laplace transform. 1   (a) (D) L1  ; s   ( s 1)(1  e ) 

  e s (b) L 1  . s   (s  5)(1  e ) 

11 Determine the transfer function of the linear system described by the following equation y( t)  5 y( t)  6 y( t)  u( t), t  0, where y(t) is the output and u(t) is the input (control) function.  N.B. Questions marked (D) will be solved during the workshop. Answers (to selected questions): 1 1 1 1 1 (a) y (t )   e  t  e 2 t  1  t ; (b) y (t )  e  3t  (5t  1)e 2t ; 5 5 3 3

(c) y( t)   t  e2t  2 te 2t  e2t ; y (t )  cos t  4e5t  8e2t ; 2 e 2 s

4s2  4s  2 ; s3

1 s 1 3 (a) (t  1)U (t  2); e 2 (  2 ) ; (b) 1  U (t  1)(t  2)  U ( t  2)(t2  t) ; s s s s 2s  2s e 1  e 2 s 1 e 2e 3e 2e 2 s ;   2  2  3 ; (d) g( t)  sin t[1 U ( t  2 )]; s s s s s s s2  1 sin 3(t  3) 4 (b) U (t  3) ; 5 (b) 3 1 1 x (t )  e t cos t  2e t sin t  [1 e 2  t (cost  sint )]U (t  2 ) [1 e4  t (cost  sint )]U (t  4 ) 2 2 3 

MATH2012EngineeringMathematics–Workshop11

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1 1 1 (d) y( t)   et  e4t  U( t  3)[3  4 et 3  e4( t 3) ] ; 3 3 6

6 (a)

1 1 1 e ; (b) sinh t  te t ;  e s 2 2 s 1 s 1

e 1 1 (c) y (t )  sinh t  te t  U (t  1)[sinh(t  1)  (t  1)e (t  1) ] 2 2 2

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