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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 difer
ential equation in the Form
ay
°°
+
by
°
+
cy
=
f
(
t
)
We know how to ±nd the general solution
y
=
y
(
t
) once we ±nd homogeneous solutions
y
1
=
y
1
(
t
) and
y
2
=
y
2
(
t
), so we explain a method to ±nd a particular solution
Y
=
Y
(
t
). ²ollow 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
°
j
=0
a
ij
t
j
e
α
i
t
cos
β
i
t
+
d
i
°
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
°°
i
+
b
(
t
)
Y
°
i
+
c
(
t
)
Y
i
=
f
i
(
t
)
For
i
=1
,
2
,...,n
;
is in the Form
Y
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 EdrayGoins

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