63559874-Unit-1-La-Place

63559874-Unit-1-La-Place - 2MARKS 1.Define the Laplace...

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Unformatted text preview: 2MARKS 1.Define the Laplace transform of a function f (t ) . The Laplace transfom of f (t ) associates afunction of S defined by the equation g J ( s ) ! ´ e  st f (t )dt 0 Here J ( s ) is said to be the Laplace transform of f (t ) and it is written as l ( f (t )) or simply l ( f ) The symbol L which transforms f (t ) into J ( s ) is called the Laplace transform operation. 2.Evaluate l ( 2t ) 2t ! elog(2 t ) ! et log 2 @ l (2t ) ! l (et log 2 ) ! 1 «Q l (e at ) ! 1/ s  a » ½ s  log 2 ­ b)Evaluate l (3t ) l (3t ) ! 1 s  log 3 l (3t ) ! l ( e log 3t ) ! l (et log 3 ) ! 1 s  log 3 3.Find l (sin 3 t ) We know that sin 3U ! 3sin U  4sin 3 U @ sin 3 U ! 3sin U  sin 3U 3sin t  sin 3t and sin 3 t ! 4 4 l (sin 3 t ) ! 1 (3l sin t  l sin 3t ) 4 1 1« 3 3» ¬ s2 1  s2  9 ¼ 4­ ½ 1 24 6 !.2 !2 2 4 ( s  1)( s  9) ( s  1)( s 2  9) ! 4.Find l sin t (2t  3) sin(2t  3) ! sin 2t.cos 3  cos 2t sin 3 @ l sin t (2t  3) ! cos 3.l (sin 2t )  l (cos 2t ).sin 3 2 s  sin 3. 2 s 4 s 4 2 cos 3  s sin 3 ! s2  4 ! cos 3. 2 5.Find the Laplace transform of t 3 l (t 3 ) ! 3! 6 !4 s 1 s 3 6.Find l (t 3 / 2  cos t  1) l (t 3 / 2  cos t  1) ! l (t 3 / 2 )  l (1)........(1) l (t 3 / 2 ) ! 3 T 4s 5 / 2 l (cos t ) ! 1 s and l (1) ! s s 1 2 Putting these in(1) we get, l (t 3 / 2  cos t  1) ! 3 s 1 T 2  5/ 2 4s s 1 s 7.Show that the Laplace transform of e5t t 3 is 6 (s  5) 4 l (e 5t t 3 ) ! [l (t 3 )]s p s  5 6 ¨ 3! ¸ !© 4 ¹ ! 4 ª s º s p s  5 ( s  5) 2 8.Show that the Laplace transform of f (t ) ! tet is 1 (s  1)2 l (tet ) ! [l (t )]s p s 1 1 ¨ 1! ¸ !© 2 ¹ ! 2 ª s ºs p s 1 ( s  1) 9.Find the Laplace transform of e2t cos2t l (e 2 t cos t ) ! [l (cost)]s p s  2 ¨s¸ !© 2 ¹ ª s  1 ºs p s  2 ! s2 s2 !2 2 ( s  2)  1 s  4 s  5 10.Find the Laplace transform of l (e 2 t sin 3t ) l (e 2t sin 3t ) ! l (sin 3t ) s p s  2 ¨3¸ !© 2 ¹ ª s  9 ºsp s 2 ¨ ¸ 3 3 !© ¹! 2 2 ª ( s  2)  9 º s  4s  13 ¨ ¸ 1 11.Find l 1 © 5¹ ª ( s  5) º ¨1 l 1 © 5 ª ( s  5) ¸ 5 t 1 ¨ 1 ¹!e l © 5 ªs º ! e5t . ! ¸ ¹ (s  5 is reduced to s) º 1 1 ¨ 4! ¸ l© ¹ 4! ª s 5 º e 5t 4 t 24 ¨ 1 ¸ at 12.Show that l 1 © ¹ ! te ( s  a) 2 º ª 3 ¨1¸ ¨1¸ l 1 © ! e at l 1 © 2 ¹ ( s  a is reduced to s) 2¹ ªs º ª ( s  a) º = eat t ! te at a ¨ ¸ 1 13.Show that l 1 © is te at 2¹ ª (s  a ) º ¨1¸ ¨1 ! e at l 1 © 2 l 1 © 2¹ ªs ª (s  a ) º ¸ ¹ (s+a is reduced to s) º = te at ¨ s ¸ is e 2t (1  2t ) 14.Show that l 1 © 2¹ ª ( s  2) º s22 s2 2 s  = = 2 2 2 (s  2) ( s  2) (s  2)2 ( s  2) ! ¨ s l 1 © 2 ª ( s  2) 1 2  s  2 (s  2)2 ¸ 1 ¨ 1 ¸ 1 1 ¹!l © ¹  2l ( s  2) 2 ª s2º º ¨1¸ ! e 2t  2e 2t .l 1 © 2 ¹ ªs º ! e 2t  2e 2t t ¨ s ¸ 2t ! e (1  2t ) l 1 © 2¹ ª ( s  2) º s3 ¸ 15.Find the inverse Laplace transform of l 1 ¨ 2 © ¹ ª s  4 s  13 º s3 s3 (s  2)  5 ! ! 2 s  4s  13 (s  2)  4  13 (s  2)2  9 2 s3 ¨ ¸ 1 ( s  2)  5 @ l 1 © 2 ¹ !l ( s  2)2  9 ª s  4 s  13 º ¨ s5 ¸ ! e 2 t l 1 © 2 ¹ ( s  2 is reduced to s) ª s 9º 4 1 ¸» «¨s¸ 1 ¨ ! e 2 t ¬ l 1 © 2 ¹  5l © 2 ¹¼ ª s  9 º½ ­ ª s 9º ! e 2t (cos 3t  5 / 2 sin 3t ) s 1 ¸ t 16.Show that l 1 ¨ 2 © ¹ ! e co s t ª s  2s  2 º s 1 s 1 ! s  2 s  2 ( s  1) 2  1 2 ¨ s  1 ¸ 1 ¨ s  1 ¸ l 1 © 2 ¹ !l © ¹ 2 ª s  2s  2 º ª ( s  1)  1 º ¨s¸ ! e  t l 1 © 2 ¹ ( s  1 is reduced to s) ª s 1 º ! e t cos t ¸ 17.Find l 1 ¨ 2 © ¹ ª s  4s  4 º 1 1 1 ! s  4s  4 (s  2)2 2 1 ¨ ¸ 1 ¨ 1 ¸ 2 t 1 ¨ 1 ¸ @ l 1 © 2 !e l © 2 ¹ ¹!l © 2¹ ª s  4s  4 º ªs º ª ( s  2) º ! e2t .t ! te2t 1 ¨ ¸ 2 t l 1 © 2 ¹ ! te ª s  4s  4 º ¨ s3 ¸ ¹ 2 ª ( s  3)  4 º 18.Find the function whose Laplace transform is © ¨ s  3 ¸ 3t  1 ¨ s ¸ l 1 © ¹!e l © 2 ¹ ( s  3 is reduced to s) 2 ªs 4º ª ( s  3)  4 º ! e3t cos 2t ¨ 1 ¸ 19.Find l 1 © 2 ¹ ª s  2s  1 º 5 1 ¨ ¸ 1 ¨ 1 ¸ l 1 © 2 ¹!l © 2¹ ª s  2s  1 º ª ( s  1) º ¨1 ! e t l 1 © 2 ªs ¸ ¹ ( s  1 is reduced to s) º ! et t 20.Using the formula on transform of derivatives ,obtain l (e  at ) Let f (t ) ! e  at €€ (1) p p Then f (t ) ! ae  at €€ (2) We know that lf d) ! slf (t )  f (0) €€ (3) (t p f (0) ! 1 €€ (4) p Substituting (1),(2),(3)and(4) we get, l (  ae  at ) ! sl (e  at )  1 sl (e  at )  gl (e  at ) ! 1 l (e  at ).( s  a) ! 1( or ) l ( e at ) ! 1 sa 21.Using the formula on transform of derivatives ,obtain l (cos at ) f (t ) ! cos at €€ (1) p Then d f d) !  a sin at €€ (2) and f d) ! a 2 cos at €€ (3) (t p (t p d d (t (0 p We know that lf d ) ! s 2lf (t )  sf (0)  f d ) €€ (4) f ( 0) ! 1 d fd)!0 (0 Substituting (1),(3),(5)and(6) in (4) we get l (  a 2 cos at ) ! s 2l (cos at )  s l (cos at ).( s 2  a 2 ) ! s (or) l (cos at ) ! s s  a2 2 22.Find the Laplace transform of (tsint) is 6 ¨1¸ We know that l (t sin t ) ! © 2 ¹ ª s 1 º d¨ 1 ¸ © ¹ ds ª s 2  1 º l (t sin t ) !  ¨ ¸ 1 ! © 2 .2 s ¹ 2 ª ( s  1) º ! 2s ( s  1)2 2 23.Find the Laplace transform of l (t 2 sin t ) We know that l (t 2 sin t ) ! l (t 2 sin t ) ! (1)2 1 s 1 2 d2 ¨ 1 ¸ © ¹ ds 2 ª s 2  1 º ¡ ! d d¨ 1 ¸ .© ¹ ds ds ª s 2  1 º ! » 1 d« ¬  ( s 2  1) 2 .2 s ¼ ds ­ ½ « (s 2  1)2 .1  s.2(s 2  1) » .2 s ¼ ! 2 ¬ ( s 2  1)4 ­ ½ ! 2 !2 ( s 2  1)2 ( s 2  1  4 s 2 ( s 2  1)4 (3s 2  1) (s 2  1)3 24.Find the inverse transform of tan 1 ( s ) Let J ( s ) ! tan 1 ( s ) (s Differentiating Jd ) ! 1 1  s2 7 ¨1 @ l 1J d ) ! l 1 © (s 2 ª 1 s ¸ ¹ ! sin t º 1 (s We know that l 1J ( s ) !  l 1J d ) t 1 !  sin t !  sin t / t t l 1 (tan 1 ( s )) !  sin t / t 25.Find the Laplace transform of sin at t g « f (t ) » We know that l ¬ ! ´ J ( s ) ds ­t¼ s ½ l (sin at ) ! a s  a2 2 g a ¨ sin at ¸ @l © ds ¹!´ 2 2 ª t º gs a g s» 1« ! a ¬ tan 1 ¼ a­ a ½s ¨s¸ ! tan 1 g  tan 1 © ¹ ªaº T ¨s¸ ¨s¸ ¨a¸ !  tan 1 © ¹ ! cot 1 © ¹ ! tan 1 © ¹ 2 ªaº ªaº ª sº 26.Find the inverse Laplace transform of e2 s / s  3 e2 s / s  3 = e2 s J ( s) where J (3) =1/s-3 p We know that l 1[e asJ ( S )] ! f (t  a ).u (t  a ) €€ (1) Where f(t)= l 1[J ( S )] l 1[1/ s  3] = e3t ! f (t ) l 1[e 2 s / s  3] ! f (t  2).u (t  2) [using formula 1] = e3t ( t  2) .u (t  2) 8 27.Define the dirac delta function: The dirac delta function is denoted by s(t-a). It exists only at t=a at t=a at which ti si infinitely great . It is difined by, s(t  a) ! 0 t{a !g g Such that ´ s(t  a )dt ! 1 g It is also called the unit impulse function. 28.State the application of impulse function: The impulse function finds applications is problems were a large force is applied for a very short time or a large force acts over a very small area as in the loading of a beam or a shock voltage applied to an electrical circuit. 29.Using the formula for Laplace transform of a periodic functions, evaluat l (cos 2t ) : cos 2t ! cos(2t  2T ) ! cos 2(t  T ) @ cos 2t is a periodic functions with period T= T T Hence l (cos 2t ) ! 1 e  st cos 2tdt 1  e  sT ´ 0 Using the formula for L.T of a periodic function, T l (cos 2t ) ! 1 e st cos 2tdt 1  e sT ´ 0 T » 1 « e  st ! s cos 2t  2sin st ¼  sT ¬ 2 1 e ­ s  4 ½0 1 1 «e  sT ( s )  ( s )» .2 !  sT ½ 1 e s 4­ (cos 2T ! 1 ! cos 0; sin 2T ! 0 ! sin 0) d (0   30.If why statisfies the equation y d 3 yd 2 y ! e t and y(0) and yd ) =0, find l ( y) . d The given equation is yd 3 yd 2 y ! e  t €€ (1)   p 9 ie( s 2  3s  2)l ( y ) ! l (e  t ) ! L( y ) ! since y(0) ! 0 ! yd ) (0 1 s 1 1 (s  1)( s  3s  2) 2 ******************* 10 11 ...
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This note was uploaded on 01/12/2012 for the course BILD 2 taught by Professor Schroeder during the Spring '08 term at UCSD.

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