$What is the transfer function of this system of differential equations?$

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$What is the transfer function of this system of differential equations?$

$2\frac{d^2y_1(t)}{dt^2} + 2\frac{dy_1(t)}{dt} + 1+2y_1(t) - 2\frac{dy_2(t)}{dt} = 0 $

$2\frac{d^2y_2(t)}{dt^2} + 2\frac{dy_2(t)}{dt} - 2\frac{dy_1(t)}{dt}=f(t)$

Where f(t) is the input
and y_2(t) is the output

I previously tried to find the transfer function by first applying Laplace transform, then substituting Y_1(s) in terms of Y_2(s) into the second equation and rearranging. The transfer function I found using this method was 4th order, had a ^-1 term in the denominator and just didn't look right to me.

If anyone can tell me what the transfer function should look for this system it would be greatly appreciated.

Ordinary Differential Equations

Isaac 210

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That $1$ in the the first equation causes problems, so I substituted $g_1(t) = 1+2y_1(t)$ and $g_2(t) = 2y_2(t)$:

$\ddot{g_1} + \dot{g_1} + g_1 - \dot{g_2} = 0$

$\ddot{g_2} + \dot{g_2} - \dot{g_1} = f(t)$

Solving for the transfer function of $\mathcal{L}\{f(t)\} = F(s)$ to $\mathcal{L}\{g_2(t)\} = G_2(s)$ yields:

$\dfrac{G_2(s)}{F(s)} = \dfrac{s^2+s+1}{s(s^3+2s^2+s+1)}$

Then, if $Y_2(s) = \mathcal{L}\{y_2(t)\}$, $G_2(s) = 2Y_2(s)$:

$\dfrac{Y_2(s)}{F(s)} = \dfrac{s^2+s+1}{2s(s^3+2s^2+s+1)}$

If this doesn't really answer your question (or if I'm wrong), feel free to email me some more of your work so I can take a closer look.

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$What is the transfer function of this system of differential equations?$

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