Linear Algebra with Differential Equations/Homogeneous Linear Differential Equations/Imaginary Eigenvalues Method

When eigenvalues become complex, mathematically, there isn't much wrong. However, in certain physical applications (like oscillations without damping) there is a problem in understanding what exactly does an imaginary answer mean? Thus there is a concerted effort to try to "mathematically hide" the complex variables in order to achieve a more approachable answer for physicists and engineers. Essentially, we have a solution that in part looks like this:

$$(\alpha + \beta \cdot i) \cdot e^{r + i \lambda_1 \cdot t}$$

But by Euler's formula:

$$(\alpha + \beta \cdot i) \cdot e^r (cos(\lambda_1 \cdot t) + i \cdot sin(\lambda_1 \cdot t))$$

Now we distribute the terms:

$$e^r(\alpha \cdot cos(\lambda_1 \cdot t) - \beta \cdot sin(\lambda_1 \cdot t)) + i \cdot e^r (\beta \cdot cos(\lambda_1 \cdot t) + \alpha \cdot sin(\lambda_1 \cdot t))$$

Since this is a linear combination of two terms, $$i$$ is a constant (complex, but still a constant), each part is an element of the set of solutions and the general solution can be constructed therein.