THE METHOD OF UNDETERMINED COEFFICIENTS
In standard mathematical terminology, “the” method of undetermined coefficients is a method
used to solve relatively simple nonhomogeneous linear differential equations with constant coef-
ficients. However, the method of undetermined coefficients for differential equations is just one
example of a family of methods used when you know the structure of a solution to a problem but
not the specific solution. In partial fraction decomposition, we write a general form of terms in
a sum, using an as yet undetermined coefficient for each. Then we do some work to reduce the
original problem to a system of simple equations for the undetermined coefficients. In differential
equations, the method of undetermined coefficients involves identifying the structure of a particular
solution to a differential equation and then using that structure to reduce the original problem to
a simple algebra problem.
The method of undetermined coefficients can be described using a large number of examples, or
it can be described in general terms using an abstract formulation. The advantage of the abstract
formulation is that once you understand it, there is very little to remember. A smaller number of
examples will suffice to illustrate the method.
Linear Differential Operators
Any linear differential equation of order nhas the general form
an(x)dny
dxn+an−1(x)dn−1y
dxn−1+· · · +a0(x)y=g(x).
All such equations can be written in the abstract form
Ly =g,
where the notation “Ly” simply refers to the left side of the differential equation.
The symbol Ltaken by itself represents a linear differential operator. These are like functions,
except that the input and output of functions are numbers or vectors, while the input and output
of linear differential operators are functions of a common variable. The notation f(x) = xcos x
says that the function fconsists of instructions to multiply the argument by its cosine. Similarly,
the notation Ly =y00 +ysays that the operator Lconsists of instructions to add the argument
function to its second derivative. The name “f” is often used to refer to an unspecified function
and only takes a particular form in examples. We’ll use the name “L” the same way. It will be a
generic function in theoretical statements and have particular forms in examples.
Operator notation has two important advantages. As noted above, it allows all linear equations
to be expressed in the same abstract notation Ly =g. The other advantage is that solving an
equation y00 +y=ex(for example) is equivalent to identifying all the functions ythat yield exas
the output of the operator L, defined in this context as instructions to add the function and its
second derivative.
The method of undetermined coefficients requires that the operator Lhave constant coefficients
and that the function gbe of a particular class. It works for y00 +y= 1, but not for y00 +xy = 1.
From here on, we’ll assume that Lis a linear differential operator of order nwith constant
coefficients.
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