Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 19
Method of Undetermined
Coefficients
We showed in theorem (22.7) that a particular solution for
ay ′′ + by ′ + cy = f(t) (19.1)
is given by
y P = 1 a er2t ∫
e (r1−r2)t (∫
)
f(t)e −r1t dt dt (19.2)
where r 1 and r 2 are roots of the characteristic equation. While this formula
will work for any function f(t) it is difficult to memorize and there
is sometimes an easier to find a particular solution. In the method of
undetermined coefficients we do this:
1. Make an educated guess on the form of y P (t) up to some unknown
constant multiple, based on the form of f(t).
2. Plug y P into (19.1).
3. Solve for unknown coefficients.
4. If there is a solution then you have made a good guess, and are done.
The method is illustrated in the following example.
Example 19.1. Find a solution to
y ′′ − y = t 2 (19.3)
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