One thing I've hated about differential equations is how I need to guess the form...

One thing I've hated about differential equations is how I need to guess the form of the solution.
e.g. it's easy to show that solutions to a linear constant-coefficient differential equation such as
$y^{‴}+2y^{″}+3y^{\prime }+4y=0$
have the form of (some linear combination of) exponentials:
You can just plug in $y=e^{ax+b}$ and show that the equation is satisfiable.
But I feel there is something wrong if I must find the solutions through guess-and-check.
Yet what I was never told, and what I seem to never be able to find from looking online, is how to derive this fact rigorously, when I don't already have the intuition necessary to guess the form of the solution?
Sitting down and working through some math, I've come up with some nonsense that works quite beautifully:
1. Place the homogeneous equation into the following matrix form:
$\overrightarrow{y}{}^{\prime }=A\overrightarrow{y}$
where, for example, we have $\overrightarrow{y}={\left[\begin{array}{ccc}y& y^{\prime }& \dots \end{array}\right]}^T$
2. Drop the arrows and pretend everything is a scalar:
$y^{\prime }=Ay$
3. Separate the, uh, variables:
$\frac{dy}{dx}=Ay$
$\frac{dy}{y}=Adx$
4. Integrate:
$\int \frac{dy}{y}=\int Adx$
$\ln y=Ax+b$
$y=e^{Ax+b}$
5. Profit!
How do I derive it properly?