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UNDETERMINED COEFFICIENTS
Method for solving a nonhomogeneous second order differential equation
This method is supposed to be simpler than the Variation of Parameters method
but is limited to equations where
f
(
x
) is a polynomial, exponential, sine or cosine.
For an alternate method see the document VariationOfParameters.pdf.
THE PROBLEM
33.
p184
y
y
x
"
sin
+
=
9
2
y
y
( )
,
'( )
0
1
0
0
=
=
THE APPROACH
To solve this problem we will find a
particular solution
y
p
that satisfies the equation and a
complementary solution
y
c
of the associated homogeneous equation
y
y
"
+
=
9
0 . Then we add the results
to obtain the general solution
y x
y
y
c
p
( )
=
+
.
Finally we apply the initial conditions to determine the
final solution.
THE PARTICULAR SOLUTION
We must search for a value
y
that can satisfy the equation.
We take a hint from the term to the right of the
equals sign,
sin2
x
.
We select a trial term which can still have this form when
its derivatives are taken:
y
A
x
B
x
p
=
+
sin
cos
2
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 Spring '08
 comech
 Equations

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