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:
¨x−4x= 3e−2t(3)
The homogeneous solutions to Equation 3are Ae2tand Be−2t, 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
