In the previous section, we learned that solutions to Linear Homogeneous Constant Coefficient (LHCC) equations often involve terms of the form
However, not every value of
will work. The correct values of
arise from solving an important algebraic equation, obtained by substituting
into the differential equation. In this section, we will focus on this process for first-order equations before building up to higher-order cases.
Let’s start with a simple first-order equation:
We want to find the value of
such that
is a solution. Substituting
into the differential equation gives:
Since
is never zero, we must have:
This tells us that
is a solution. Therefore, the general solution is:
The equation
which gave us the value of
is called the
characteristic equation. Every LHCC equation has one. The characteristic equation gives us the values of
that we need to construct the general solution.
Applying the same approach to the general first-order LHCC equation gives us:
This results in the characteristic equation and general solution:
Now, let’s apply this method to a couple of examples.
This straightforward method works for any first-order LHCC equation by using the characteristic equation. In the next sections, we’ll extend this technique to higher-order equations.