Method of undetermined coefficients
Igors Gorbovickis
November 15, 2009
The method of undetermined coefficients helps to find a particular so
lution of a linear
nonhomogeneous
differential equation provided that the
equation satisfies some certain conditions.
Definition 1.
A particular solution is just any solution of a given differential
equation.
When applicable?
When the equation has
constant coefficients
(the coefficients do not depend
on
x
), and the right part of the equation is the sum of products of polynomials
and functions of the form
e
ax
,
cos(
bx
)
and
sin(
cx
)
.
Example 2.
The method is applicable to the equation
4
y
+
y

e
π
y
=
x
+ 2 sin
2
(3
x
) +
x
2
e
6
x
cos(2
x
) sin(7
x
)
,
and is not applicable to the equations
y
+
xy

y
= cos(
πx
)
and
y

2
y
+ 2
y
=
e
sin
x
.
The method
Step 1: Solving the corresponding homogeneous equation
We solve the same equation but with the zero right part (the corresponding
homogeneous equation). That equation can be solved by using the charac
teristic equation. The general solution of the homogeneous equation is called
the
complementary solution
of the original equation.
1
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Example 3.
y
+ 4
y
= 0
.
The characteristic equation is
r
2
+ 4 = 0
. The roots are
r
=
±
2
i
, and
y
c
=
A
cos(2
x
) +
B
sin(2
x
)
.
Remark 4.
We can obtain a general solution of a linear nonhomogeneous equation
by taking its complementary solution and adding any particular solution to it. Thus
the method of undetermined coefficients actually gives us all solutions of the original
equation.
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 '07
 TERRELL,R
 Differential Equations, Trigonometry, Equations, Trigonometric Identities, Sin, Cos, Euler's formula

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