Recently I have been improving my skills in Linear differential equations and I came across a problem with a rather problematic solution.
The problem is as follows: (Already converted from y prime form, I assure without error, I have checked five times)
$D^2+D+1=0$
The solution provided is:
$y = (C_{1}\cos(\frac{\sqrt{3}}{2})x+C_{2}\sin(\frac{\sqrt{3}}{2})x)\cdot e^{-x/2}$
It is clear to me why sin and cos are present, but seems to me this solution implies
$\lambda = a + bi = -\frac{1}{2} + \frac{\sqrt{3}}{2}$
Assuming,
$y_1 = e^{ax}\cos(bx), y_2 = e^{ax}\sin(bx)$.
I attempted a solution using the result obtained from the answers provided (and I believe it is not a typo as there are similar problems with similar solutions in the text I am using). The issue that I have run against is that upon attempting the imaginary factorization I cannot obtain a similar solution.
$D^2+D+1=0$
$(D+[-\frac{1}{2} + \frac{\sqrt{3}}{2}i])^2 = 0$
$D^2 - D + \sqrt{3} D i - \frac{1}{2} - \frac{\sqrt{3}}{2} = 0$
As is written, the expansion that I perform results in an expression different from the original. I assume I am making an error somewhere but I can't seem to find it and I've spent quite some time trying to figure out why.
I am sure it is some simple error that I have made and I would be extremely grateful if someone would be willing to take the time to point it out.
Thank You,
David
