Week 7:
2. 4.3 Undetermined Coef [Tables]
TABLE of trial
y
p
:
Usual trial
y
p
F
(
x
)
Usual
Modified
ce
ax
y
p
=
A
0
e
ax
ce
ax
cos(
bx
) or
y
p
=
e
ax
(
A
0
cos(
bx
)+
ce
ax
sin(
bx
)
B
0
sin(
bx
))
cx
k
y
p
=
A
0
+
A
1
x
+
· · ·
+
A
k
x
k
When:
The root associated with
F
(
x
) is NOT
a root of the characteristic polynomial
P
(
r
) giving
y
H
in the general solution
y
=
y
H
+
y
p
.
In the 3 cases,
P
(
a
) = 0;
P
(
a
+
bi
) = 0; and
P
(0) = 0
.
Applications often feature the “pure imaginary” complex
case, where
a
= 0
,
for which we get the usual solution
when
P
(
ib
) = 0
.
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2
TABLE of trial
y
p
:
Simple modified solution
F
(
x
)
Usual
Modified
ce
ax
y
p
=
A
0
e
ax
y
p
=
A
0
xe
ax
ce
ax
cos(
bx
) or
y
p
=
e
ax
(
A
0
cos(
bx
)+
y
p
=
xe
ax
(
. . .
ce
ax
sin(
bx
)
B
0
sin(
bx
))
cx
k
y
p
=
A
0
+
A
1
x
+
· · ·
+
A
k
x
k
y
p
=
x
(
. . .
When:
The root associated with
F
(
x
) is a simple
root of the characteristic polynomial
P
(
r
)
.
If the DE w
y
+
a
1
y
+
a
2
y
=
F,
so
P
(
r
) =
r
2
+
a
1
r
+
a
2
,
the cases
are (1)
P
(
r
) = 0 has distinct real roots, one of which is
r
1
=
a
; (2)
P
(
r
) = 0 has complex roots
a
±
bi,
the
SAME complex roots as the root associated with
F
(
x
)
.
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 Spring '08
 zhang
 Linear Algebra, Algebra, Complex number, yp, trial yp, Usual yp, ceax Usual yp

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