Differential Equations
Second
Quarter SY 0506
____________________________________________________________________________________________________________
Silva, Alviso and Llacuna
– Mathematics Department, MIT
133
Lesson 27
The Method of Undetermined Coefficients
Specific Objectives:
At the end of the lesson, the students are expected to:
•
differentiate the different methods on the determination of the particular
solution
•
solve non homogeneous equations using different methods
A nonhomogeneous linear differential equation with constant coefficients is of the form:
( ) ( )
−
−
−
+
+
+
+
+
=
1
2
0
1
2
1
...
n
n
n
n
n
a D
a D
a D
a D a y
R x
To find its general solution, y = y
c
+ y
p
. The complementary solution, y
c
, may be
determined from the roots of the auxiliary equation, f(m) = 0, and the particular solution,
y
p
, may be determined by:
The particular solution may be found by any of the following methods:
1.
The Method of Undetermined Coefficients
2.
Variation of Parameters
3.
Inverse Operators
4.
By Inspection
1. The Method of Undetermined Coefficients
This method may be used if R(x) is itself a solution of some homogeneous linear
differential equation with constant coefficients
.
Step 1.
Consider the right hand side of the equation, R(x) and determine its
roots.
Case 1.
If there is no repetition of roots between the auxiliary
equation and R(x) such that
when R(x) is a
then y
P
is
a. constant like 2, 3, etc.
A
b. linear function like 3x, 2x + 5, etc.
Ax + B
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Second
Quarter SY 0506
____________________________________________________________________________________________________________
Silva, Alviso and Llacuna
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 Spring '11
 DanteSilva
 Differential Equations, Equations, Sin, Cos, Quarter SY, Llacuna – Mathematics Department

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