# pe.7 write a system in the matrix form Y'=AY+G

#### karush

##### Well-known member
consider the non-homogeneous first order differential system
where $x,y,z$ are all functions of the variable t
\begin{align*}\displaystyle
x'&=-4x-3y+3z\\
y'&=3x+2y-3z+e^t\\
z´&=-3x-3y+2z
\end{align*}
write a system in the matrix form $Y'=AY+G$

#### Opalg

##### MHB Oldtimer
Staff member
consider the non-homogeneous first order differential system
where $x,y,z$ are all functions of the variable t
\begin{align*}\displaystyle
x'&=-4x-3y+3z\\
y'&=3x+2y-3z+e^t\\
z´&=-3x-3y+2z
\end{align*}
write a system in the matrix form $Y'=AY+G$
Start by taking $Y = \begin{bmatrix}x\\y\\z\end{bmatrix}$. Then the equation $Y'=AY+G$ becomes $$\def\dot{\phantom{\bullet}} \begin{bmatrix}x'\\y'\\z'\end{bmatrix} = \begin{bmatrix}\dot&\dot&\dot\\ \dot&\dot&\dot \\ \dot&\dot&\dot\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} + \begin{bmatrix}\dot\\ \dot\\ \dot\end{bmatrix}.$$ Can you fill in the blanks?

#### karush

##### Well-known member
$$Y'=\left[\begin{array}{rrrr}-4&-3&3\\3&2&-3&+e^t\\ -3&-3&2\end{array}\right] \begin{bmatrix}x\\y\\z\end{bmatrix} + \begin{bmatrix}G_1\\ G_2\\G_3\end{bmatrix}$$

ok I don't know if this is all they want but presume what we do next is a rref on A
also this has an $e^t$ in it.

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