When solving a non-homogeneous linear differential equation, selecting the correct form for the particular solution,
is just the beginning. The next step is determining the values of its unknown coefficients, such as
and
In this section, we’ll explore how to compute these coefficients by substituting the guessed particular solution into the original equation.
Let’s revisit the example we discussed earlier. Consider the differential equation:
We guessed that the particular solution would be linear, so we set
Our goal now is to determine the values of
and
To do this, we substitute
into the original differential equation. Here’s how it works:
Substituting
and
into
(38) results in:
By matching the coefficients of like terms on opposite sides of the equation,
we solve:
Thus, the particular solution is:
Regardless of
’s form, the process of finding the coefficients remains consistent:
Substitute the chosen into the differential equation.
Collect and match coefficients of like terms on both sides of the equation.
Solve the resulting system to find
This is why the method is called the "method of undetermined coefficients." and the complete process is summarized as follows.
Now for a few examples to illustrate the complete application of the nndetermined coefficients method.