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I'm asked to find a 2nd order linear homogeneous differential equation with constant coefficients, given a solution.

This is my work:

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    That's way more complicated than you need. You should be able to look at that equation and see that you just need to find a quadratic polynomial with roots $-1 \pm 3i$. – 2017-02-27
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    To discover where an error is: try plugging in the formula for $y$ to some line in the middle and see if it is correct. Then you will know whether your error is before this point of after it. Then narrow down, plugging in to a line somewhere between a correct line and an incorrect line. – 2017-02-27
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    You forgot to differentiate the $y$ when you calculated $y^{''}$ ... Ian is right there is an easier method – 2017-02-27
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    Oh I realize that now. The problem was I first calculated y'' before actually noticing y was included in the equation for y' – 2017-02-27

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Given that problem, I would notice that the solution is of the form $y=e^{ax}(c_{1}\sin bx + c_{2}\cos bx)$

Where $a\pm bi$ are the roots to the characteristic equation of the 2nd order ODE, with $a = -1, b = 3$.

So if our 2nd order ODE is of the form $y'' + Ay' + C = 0$, then it is simple to prove that:

$(-1+3i)(-1-3i) = C \Rightarrow C = 10$

$-(-1+3i -1 -3i) = A \Rightarrow A = 2$

Then the second order ODE is:

$y'' +2y'+10=0$

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    I can see it's a much easier method, and I thank you for showing it to me; however, the problem says I have to find the DE by finding the second derivative of the given solution and eliminating the Constants 1 and 2. – 2017-02-27
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    Okay, but then it would have been helpful to include that information when you asked the question. Anyways I've looked at your method and found your mistake: when you differentiate $y'$, you forgot to include the y term – 2017-02-27
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    Yes, I see it now. It's my first post but you're right, and I'll keep that in mind next time. Thanks for your help. – 2017-02-27
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    @Sam202 If this helped you, please [accept this answer](https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work). That way, it lets others know that your problem is solved. Additionally, it gives you +2 reputation. – 2017-04-05