Dealing with complex roots and more on independence -...

Dealing with complex roots and
more on independence

Complex roots
• When the roots of the auxiliary equation for
the differential equation ay’’+by’+cy=0
ar
2
+br+c=0
are complex ( b
2
‐ 4ac<0 ) , they come as a
complex conjugate pair
r= α ±i β
• In that case we have (complex) solutions
y
h
=C
1
e
( α +i β )t
+C
2
e
( α‐ i β )t
=e
α t
(C
1
e
i β t
+C
2
e
‐ i β t
)

Complex roots
• y
h
=e α t (C
1
e i β t +C
2
e ‐ i β t )
=e
α t
(C
1
( cos β t +isin β t) +C
2
( cos β t ‐ isin β t) )
=e
α t
((C
1
+C
2
) cos β t + (iC
1
‐ iC
2
) sin β t
=e
α t
(D
1
cos β t +D
2
sin β t )
since the C’s were arbitrary constants, we can
reformulate and use D’s. This form has the
advantage of producing real values when D’s
are real.

Repeated roots
• In the case with one real root, (b 2 =4ac ) we can
seek a second independent solution in the
form y=ue
rt .
In this case, we know that
r= ‐ b/2a
• Plugging in to the differential equation we get
a(u”+2ru’+r
2
u)e
rt
+b(u’+ru)e
rt
+cue
rt
=0
rearranging we get
((ar
2
+br+c)u+(2ar+b)u’+au’’)e
rt
=0