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I had a system of differential equations and I ended up in this.

$4x+2y+4z=0$ $2x+y+2z=0$ $4x+2y+4z=0$

I don't know what to do from here.

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    Would you know what to do if you ended up with just one equation, $2x+y+2z=0$?2012-01-26

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Note that your system of equations is actually a single equation in various avatars. (i.e.) one is scalar multiple of the other. Put another way, these equations convey the same information, nothing new!

$4x+2y+4z=0 \iff 2(2x+y+2z=0) \iff 4x+2y+4z=0$ Since there are three unknowns, and only one equations, two parameters are required to fix a solution. We can have a parametric solution as follows:

Set $x=\lambda$ and $y=\mu$. Then, $2\lambda+\mu+2z=0 \implies z=\dfrac{-2\lambda-\mu}{2}$

Hence, the triplet $(\lambda,\mu,\dfrac{-2\lambda-\mu}{2})$ with $\lambda,\mu \in \mathbb R$ is a solution to your system.

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    That was a typo! I am sorry. I fixed it now!2012-01-26