# Solve the following differential equation: $y'' + y' + y = 0$

Alright, so this is what I've done so far:

1. First, form the characteristic quadratic equation, and solve it.
2. I solved it using the quadratic formula to get solutions

$-\frac{1}{2} \pm i\frac{\sqrt{3}}{2}$

1. Then the answer should be $y(x) = e^{x/2}(A\cos(\sqrt{3}x/2) + B\sin(\sqrt{3}x/2))$ right?
2. Webworks gives me two blanks to fill in: It has $y(x) = C_1$ ____ + $C_2$ _____ where $C_1$ and $C_2$ are constants.
3. I just distributed the $e^{x/2}$ and put in $e^{x/2}\cos(\sqrt{3}x/2)$ and the same thing except replacing $\cos x$ with $\sin x$. I also tried swapping the answers. Also, I found out it doesn't accept answers with the imaginary unit in them.
4. Any idea what format my professor could want the answers in? Am I even doing the problem correctly?

Any help is greatly appreciated! I'll respond quickly

The comment below just solved my problem I believe, thank you!

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You have stated what the solution to the differential equation is in step 3. The solution (after evaluating the quadratic formula correctly!) is just fine. At my ODEs class that would be accepted. –  user38268 Aug 4 '11 at 2:57
^^^ I was really close to being correct, J. Mohr found out the problem. The e^(x/2) needs to be changed to e^(-x/2). Oh negative signs, how I love you –  user13327 Aug 4 '11 at 3:02

Since the roots of characteristic equation are $-\frac{1}{2} \pm i\frac{\sqrt{3}}{2}$, the general solution in WeBWork-friendly form is $$y(x) = C_1 e^{-x/2}\cos(\sqrt{3}x/2) + C_2 e^{-x/2}\sin(\sqrt{3}x/2))$$ (Getting the question off the unanswered list.)