MA 36600 LECTURE NOTES: MONDAY, MARCH 2
Method of Undetermined Coefficients
Undetermined Coefficients.
Say that we wish to solve a
constant coefficient
linear second order differ
ential equation in the form
a y
00
+
b y
0
+
c y
=
f
(
t
)
We know how to find the general solution
y
=
y
(
t
) once we find homogeneous solutions
y
1
=
y
1
(
t
) and
y
2
=
y
2
(
t
), so we explain a method to find a particular solution
Y
=
Y
(
t
). Follow these three steps:
#1. Express the function on the righthand side as the sum of functions
f
(
t
) =
f
1
(
t
) +
f
2
(
t
) +
· · ·
+
f
n
(
t
)
where each
f
i
(
t
) is the product of a polynomial, an exponential function, and a trigonometric func
tion. That is, say that we can write
f
i
(
t
) =
d
i
X
j
=0
a
ij
t
j
e
α
i
t
cos
β
i
t
+
d
i
X
j
=0
b
ij
t
j
e
α
i
t
sin
β
i
t
for some constants
a
ij
,
b
ij
,
α
i
, and
β
i
. Note that
f
i
(
t
) involves a polynomial of degree
d
i
.
#2. Make a guess that a solution
Y
i
=
Y
i
(
t
) of the nonhomogeneous equation
a
(
t
)
Y
00
i
+
b
(
t
)
Y
0
i
+
c
(
t
)
Y
i
=
f
i
(
t
)
for
i
= 1
,
2
, . . . , n
;
is in the form
Y
i
(
t
) =
d
i
+2
X
j
=0
A
ij
t
j
e
α
i
t
cos
β
i
t
+
d
i
+2
X
j
=0
B
ij
t
j
e
α
i
t
sin
β
i
t
for some constants
A
ij
and
B
ij
. Note that
α
i
, and
β
i
are the same as above, and that
Y
i
(
t
) involves
a polynomial of degree (
d
i
+ 2).
#3. Recombine as the sum
Y
(
t
) =
Y
1
(
t
) +
Y
2
(
t
) +
· · ·
+
Y
n
(
t
);
then
Y
=
Y
(
t
) is the desired solution to the nonhomogeneous equation
a Y
00
+
b Y
0
+
c Y
=
f
(
t
)
.
This is known as the
Method of Undetermined Coefficients
.
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 Spring '09
 MA
 Derivative, Continuous function, Yi, general solution

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