Partial fractions 19
x^3 + 4 x^2 + 20 x− 7
(x− 1 )^2 (x^2 + 8 )
[
3
(x− 1 )+2
(x− 1 )^2+1 − 2 x
(x^2 + 8 )]- When solving the differential equation
d^2 θ
dt^2
− 6dθ
dt− 10 θ= 20 −e^2 t by Laplace
transforms, for given boundaryconditions,thefollowing expression forL{θ}results:L{θ}=4 s^3 −39
2s^2 + 42 s− 40
s(s− 2 )(s^2 − 6 s+ 10 )
Show that the expression can be resolved into
partial fractions to give:L{θ}=2
s−1
2 (s− 2 )+5 s− 3
2 (s^2 − 6 s+ 10 )