I have a somewhat theoretical question regarding Differential Equations:(adsbygoogle = window.adsbygoogle || []).push({});

How can we reconcile the fact that if I go from let's say this system of 1st ODE

x' = 2y-x

y' = -x+y

to a 2nd ODE "using x(t) instead of y(t)" we get: x" + x =0

then back to a system of 1st ODE:

letting y=x' and then from x"=-x we get x"=-y' so y'=-x.

So now our new 1st ODE system has the following equations instead

x'=y

y'=-x

which is different from our original system:

x' = 2y-x

y' = -x+y

Is there a good Mathematical explanation to this?

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# From a System of 1st ODE to a 2nd ODE and back to the system of 1st ODEs

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