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Method of Undetermined Coefficients: Determining Particular Solutions of Linear ODEs, Study notes of Differential Equations

The method of undetermined coefficients, a technique used to find particular solutions of linear ordinary differential equations (odes) with constant coefficients when the forcing function has a finite number of linearly independent derivatives. Examples of common forcing function and solution pairs and discusses complications that arise when the forcing function is proportional to a homogeneous solution. It also mentions the annihilator operator theory as a systematic approach to dealing with such situations.

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Method of Undetermined Coefficients
D. S. Stutts
January 12, 2006
The method of undetermined coefficients is used to determine the particular solution of linear ordinary
differential equations with constant coefficients when the forcing function has a finite number of linearly
independent derivatives.
For example, given an ODE of the form:
¨x+α˙x+βx =f(t),(1)
where αand βare constants, we seek particular or forced solutions proportional to f(t) and all of its
derivatives hence the need for a finite number of them!
Table 1. Forcing function and matching solution form.
Forcing Function Particular Solution
2t3At3+Bt2+C
10 sin(3t)Acos(3t) + Bsin(3t)
4eat Aeat
2 A
Luckily, the forcing function and solution pairs shown in Table 1are fairly commonly encountered.
However, the vast majority of possible functions do not have a finite number of linearly independent
derivatives. For example:
cf(t) = 1
t(2)
˙
f(t) =
1
t2
¨
f(t) = 2
t3
.
.
.
f(n)(t)=(1)nn!
tn+1
Another complication arises when the forcing function is proportional to a homogeneous solution. For
example:
¨x4x= 3e2t(3)
The homogeneous solutions to Equation 3are Ae2tand Be2t, so we must make the particular solution
independent of the homogeneous solutions. This is accomplished by multiplying the repeated eigenfunction
(homogeneous solution) by the independent variable, t. For the above example, the particular solution is
given by:
1
pf2

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Method of Undetermined Coefficients

D. S. Stutts

January 12, 2006

The method of undetermined coefficients is used to determine the particular solution of linear ordinary differential equations with constant coefficients when the forcing function has a finite number of linearly independent derivatives.

For example, given an ODE of the form:

x¨ + α x˙ + βx = f (t), (1) where α and β are constants, we seek particular or forced solutions proportional to f (t) and all of its derivatives – hence the need for a finite number of them!

Table 1. Forcing function and matching solution form.

Forcing Function Particular Solution 2 t^3 At^3 + Bt^2 + C 10 sin(3t) A cos(3t) + B sin(3t) 4 eat^ Aeat 2 A

Luckily, the forcing function and solution pairs shown in Table 1 are fairly commonly encountered. However, the vast majority of possible functions do not have a finite number of linearly independent derivatives. For example:

cf (t) =

t

f˙ (t) = − 1 t^2 f^ ¨ (t) =^2 t^3 .. . f (n)(t) = (−1)n^ n! tn+ Another complication arises when the forcing function is proportional to a homogeneous solution. For example: ¨x − 4 x = 3e−^2 t^ (3) The homogeneous solutions to Equation 3 are Ae^2 t^ and Be−^2 t, so we must make the particular solution independent of the homogeneous solutions. This is accomplished by multiplying the repeated eigenfunction (homogeneous solution) by the independent variable, t. For the above example, the particular solution is given by:

xp(t) = −

te−^2 t^ (4)

A systematic approach to dealing with the above situation is found in the so-called annihilator operator theory – a discussion of which may be found in any text on elementary ordinary differential equations.