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.