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Finding a particular solution for a constant coefficient linear ODE (enrichment)
Linear nonhomogeneous ODEs with constant coefficients can be solved utilizing the principle of
superposition. For example, the general solution to the ODE below can be found as a sum of the
general solution to the corresponding homogeneous ODE and any function f(x) that satisfies the ODE,
i.e., the particular solution.
)
(
2
2
x
g
cy
dx
dy
b
dx
y
d
a
=
+
+
Equation (1)
here
a, b, c
= constants
p
ogeneous
y
y
x
y
+
=
hom
)
(
While finding the homogeneous solution is relatively straightforward, finding the particular
solution may be challenging, especially for higher order ODEs. The following two methods are
commonly used to find the particular solution.
1) Method of undetermined coefficients.
With this method, we “guess” the form of the particular solution based on the form of g(x),
with undetermined coefficients to be solved for by substituting the guessed form of the
solution into the ODE. For example,
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This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.
 Spring '10
 SerhiyYarusevych

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