determine the following functions (v/g)x

v= x² + 5x+4 g= x² + 2x-8

Princess Erika Vegafria
2022-10-04

determine the following functions (v/g)x

v= x² + 5x+4 g= x² + 2x-8

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What is the frequency response of a first-order system of ODEs?

What is the frequency response of a first-order system of ODEs? Specifically, given the differential equation:

${y}^{\prime}\left(t\right)=Ay\left(t\right)+x\left(t\right)$

where y: $y:[0,\mathrm{\infty}]\to {\mathbb{R}}^{m}$ and $A\in {\mathbb{R}}^{m\times m}$, what is the solution when

$x\left(t\right)=\left[\begin{array}{c}0\\ \vdots \\ \mathrm{cos}\left(\omega t\right)\\ \vdots \\ 0\end{array}\right]=\mathrm{cos}\left(\omega t\right){b}_{i}?$

Essentially, if we have a first-order system of ODEs and we activate a single input with a sine wave of a given frequency, what is the solution? Normally, the frequency response is worked out for a linear time invariant system of higher order and the solution is just a phase shifted and amplitude scaled version of the sine wave. I'm interested in what this result looks like for a first-order system rather than a single equation.

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or each $({x}_{0},{y}_{0})\in \mathbb{R}\times \mathbb{R}$ I need to prove that there is a single solution defined on $\mathbb{R}$

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