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

${y}^{\u2034}+2{y}^{\u2033}+3{y}^{\prime}+4y=0$

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:

$\overrightarrow{y}{\phantom{\rule{thinmathspace}{0ex}}}^{\prime}=A\overrightarrow{y}$

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:

$\frac{dy}{dx}=Ay$

$\frac{dy}{y}=A\phantom{\rule{thinmathspace}{0ex}}dx$

4. Integrate:

$\int \frac{dy}{y}=\int A\phantom{\rule{thinmathspace}{0ex}}dx$

$\mathrm{ln}y=Ax+b$

$y={e}^{Ax+b}$

5. Profit!

How do I derive it properly?