How to represent it as a matrix?
\begin{align} \dfrac{dx}{dt}&=4x+4y-24, \\ \dfrac{dy}{dt}&=-8x+16y+60. \end{align}
Or, at least, how to represent a single equation:
\begin{align} \dfrac{dx}{dt}&=4x+4y-24, \end{align}
?
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$$\begin{pmatrix}\dot x\\\dot y\end{pmatrix}=\begin{pmatrix}4&4\\-8&16\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}-24\\60\end{pmatrix}$$
Surb
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I can add 2x2 matrix to 2x1 matrix? – Stdugnd4ikbd Feb 19 '19 at 18:12
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$\begin{pmatrix}4&4\-8&16\end{pmatrix}\begin{pmatrix}x\ y\end{pmatrix}$ is $2\times 1$ matrix. – mfl Feb 19 '19 at 18:13
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@mfl, thanks. . – Stdugnd4ikbd Feb 19 '19 at 18:14
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Off topic, but why you divided exactly $dy$ by $dx$, but not vice versa? – Stdugnd4ikbd Feb 19 '19 at 18:15
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@АртурКлочко: You can also compute $dx/dy$. But neither of these equation gives an easy ODE to solve. – Surb Feb 19 '19 at 18:23
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Ok, and If I want to get the eigenvalues of this system to determine type of the singularity point, considering the fact, that this system is a linearization of nonlinear system, near some point, what I have to do? – Stdugnd4ikbd Feb 19 '19 at 18:27
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Just compute the eigenvalue of $\begin{pmatrix}4&4\-8&16\end{pmatrix}$ ;-) – Surb Feb 19 '19 at 18:34
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Oh I thought You didn't answer. But as I read, to find an eigenvalues of nonhomogeneous system, I need, at first find a homogeneous solution, then add a nonhomogeneous solution, or something like that, no? I honestly, made a research, I have read a lot of articles, but I didn't understand.. – Stdugnd4ikbd Feb 20 '19 at 12:03
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That's all, I understood, thank You – Stdugnd4ikbd Feb 20 '19 at 12:50
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It is a non-homogeneous linear system:
$$\begin{pmatrix}\dot x\\ \dot y\end{pmatrix}=\begin{pmatrix}4&4\\ -8&16\end{pmatrix}\begin{pmatrix} x\\ y\end{pmatrix}+\begin{pmatrix}-24\\ 60\end{pmatrix}.$$
Bernard
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If I want to get the eigenvalues of this system to determine type of the singularity point, considering the fact, that this system is a linearization of nonlinear system, near some point, what I have to do? – Stdugnd4ikbd Feb 19 '19 at 18:20