First-Degree Homogeneous Linear Equations with Constant Coefficients
Now that we solved one differential equation, let’s try a wider class: “first-degree homogeneous linear equations with constant coefficients”. Let’s break this down.
- First-degree: only involves the undetermined function and its first derivative
- Homogeneous: harder to nail down, but for our purposes it means that every term involves the undetermined function or its first derivative
- Linear: no products of the undetermined function or its first derivative with each other
- Constant Coefficients: only multiplying the undetermined function and its derivative by constants
Putting it all together, this means that our equation must look like this:
We can divide through by to assume without loss of generality that (if then the equation isn’t very interesting at all).
Now let’s again assume that is analytic at , so we can write
So our equation reads
That is, for all , and is arbitrary. Just like last time we see that multiplying by at each step gives a factor of . But now we also multiply by at each step, so we find
And indeed, we can rewrite our equation as . The chain rule clearly shows us that satisfies this equation.
In fact, we can immediately see that the function will satisfy many other equations, like , , and so on.