Matematicas

1. Calculeu la Transformada de Laplace de les funcions següents:
1.1. f (t ) = 6 +

14
t − 4 sin 2t
8

1.2. f (t ) =

3
cos(4t ) − 6t 8 +4
2

1.3. f (t ) =

1 −8 t 5
e t + e 2t u (t + 3)
10

1.4. f (t ) = e t u (t + 22) + 2e −8t sin( 4t )

1.5. f (t ) =

1
(t − 7) 3 u(t − 7)
2

1.6. f (t ) = 3 sin(t − 4)u (t − 4)
SOLUCIONS

1.1. L( f (t )) = 6 L(1) +

1
6 1 4!
2
63
8
L(t 4 ) − 4 L(sin 2t ) = +
−4 2= + 5− 2
5
8
s 8s
s +4 s s
s +4

Per la linealitat
1.2. L( f (t )) =

1.3.

3
3s
8! 4
L(cos(4t )) − 6 L(t 8 ) + 4 L(1) =
−6 9 +
22
2 s + 16
s
s

L( f (t )) =

[

Per la linealitat

=

]

1
1
L(e −8t t 5 ) + L(e 2 t u (t + 3)) =
L (t 5 ) s = s +8 + [L (u (t +3))]s = s − 2
10
10
Per la translació complexa

 e 3s 
1  5! 
1
e 3( s − 2 )
+ 
= 12
+
10  s 6  s = s +8  s  s = s −2
s−2
(s + 8) 6


1.4. L ( f (t )) = L (e u (t + 22)) + 2 L (e
t

Per la linealitat

−8 t

sin( 4t )) = [L(u (t + 22))]s = s −1 + 2[L(sin( 4t))]s = s +8

Per la translació complexa

 e 22 s 
e 22 ( s −1)
8
4
+ 2 2
=
+
=


s − 1 ( s + 8) 2 + 16
 s + 16  s = s +8
s  s = s −1

1

1
1 −7 s 3
1 −7 s 3! 3e −7 s
3
1.5. L( f (t )) = L((t − 7) u (t − 7)) = e L(t u (t )) = e
=4
2
2
2
s4
s

Per lalinealitat

Per la translació temporal

1.6. L( f (t )) = 3L(sin(t − 4)u (t − 4)) = 3e

−4 s

3e −4 s
L(sin(t )u (t )) = 2
s +1

2