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# 4: Linear Systems of Ordinary Differential Equations (LSODE)

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Definition of Linear Systems of Ordinary Differential Equations (LSODE)

\begin{eqnarray*}
y_{1}'(t)& =& \alpha_{11}(t)y_{1}(t)\ +\alpha_{12}(t)y_{2}+\ldots+\alpha_{1n}(t)y_{n}+f_{1}(t)\\
y_{2}'(t)& =& \alpha_{21}(t)y_{1}(t)\ +\alpha_{22}(t)y_{2}+\ldots+\alpha_{2n}(t)y_{n}+f_{2}(t)\\
y_{n}'(t)& =& \alpha_{n1}(t)y_{1}(t)\ +\alpha_{n2}(t)y_{2}+\ldots+\alpha_{nn}(t)y_{n}+f_{n}(t),\\
\end{eqnarray*}

where $$y_{i}'(t) :=\displaystyle \frac{dy_{i}(t)}{dt}$$ denotes the first derivative of functions $$y_{i}(t),\ i= 1,2, \ldots,\ n$$, with respect to $$t$$.

Matrix Form of LSODE

$$\lfloor_{y_{n}'(t)}|^{y_{2}'(t)}\lceil 1\ldots]=\lfloor\alpha_{n1}(t)|^{\alpha_{21}.(t)}\lceil\alpha_{11}.(t)\alpha_{n2}(t)\alpha_{22}.(t)\alpha_{12}.(t).\cdot.\cdot.\ \alpha_{nn}(t)\alpha_{2n}.(t)\alpha_{1n}.(t)]\lfloor y_{n}(t)|^{y_{2}(t)}\lceil y_{1}.(t)]+\lfloor_{f_{n}(t)}|^{f_{2}(t)}\lceil f_{1}.(t)]$$,

or in matrix form

\begin{eqnarray*}
y^{\prec}(t)=A(t)\vec{y}(t)+f^{\rightarrow}(t),
\end{eqnarray*}

where

$$\vec{y}(t)=|\lceil y_{2}'(ty_{1}'(t)\rceil_{1}, A(t) =|\lceil\alpha_{21}.(t\alpha_{11}.(t) \alpha_{22}.(t\alpha_{12}.(t) \alpha_{2n}.(t\alpha_{1n}.(t)\rceil_{1}, f^{\rightarrow}(t)=|\lceil f_{2}(tf_{1}(t)\rceil_{1} \lfloor y_{n}'(t) \rfloor \lfloor\alpha_{n1}(t) \alpha_{n2}(t) \alpha_{nn}(t) \rfloor \lfloor f_{n}(t) \rfloor$$, (3), $$A(t)$$in (3) is called coefficient matrix of (2) and $$f^{\rightarrow}(t)$$