082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

658 Chapter 21 The Laplace transform
These simultaneous algebraic equations need to be solved for X(s) and Y(s). By
Cramer’s rule(see Section 8.7.2)
∣ s +11/s
s/2 0 ∣ −1/2
Y(s)=
∣ s +1 s/2
=
(s+1) 2 −s 2 /4
s/2 s+1∣
= − 1 { }
1
2 3s 2 /4+2s+1
= − 1 { }
4
2 3s 2 +8s+4
= − 1 { }
4
2 (3s +2)(s +2)
Using partialfractions wefind
Y(s)=− 1 { 3
2 3s+2 − 1 }
s +2
= − 1 { 1
− 1 }
2 s + 2 s +2
3
and hence
Similarly,
and so
y(t) = 1 2 (e−2t −e −2t/3 )
X(s)=
4(s +1)
s(3s +2)(s +2) = 1 s − 1
2 ( 1
) −
s + 2 2(s +2)
3
x(t)=1− 1 2 (e−2t/3 +e −2t )
EXERCISES21.10
1 UseLaplace transformsto solve
(a) x ′ +x=3,
x(0) =1
(b) 3 dx
dt +4x=2,
x(0) =2
(c) 2 dy
dt +4y=1,
y(0) =4
(d) 4 dy
dt +8y=7,
y(0) =6
(e) y ′′ +y ′ +y=1,
y(0) =1,y ′ (0) =3
2 UseLaplacetransformsto solve
(a) x ′′ +x =2t,
x(0) =0,x ′ (0) =5
(b) 2x ′′ +x ′ −x =27cos2t +6sin2t,
x(0) = −1,x ′ (0) = −2