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

In summary, the conversation discusses a non-homogeneous first order differential system with variables $x, y, z$ and a matrix form equation $Y'=AY+G$. The matrix A is partially filled in and the next step is to perform a rref on it. The system also contains an $e^t$ term.
  • #1
karush
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consider th 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 $Y'=AY+G$
 
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  • #2
karush said:
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?
 
  • #3
$$Y'=\left[\begin{array}{rrrr}-4&-3&3\\3&2&-3&+e^t\\ -3&-3&2\end{array}\right]
\begin{bmatrix}x\\y\\z\end{bmatrix}
+ \begin{bmatrix}G_1\\ G_2\\G_3\end{bmatrix}$$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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FAQ: -pe.7 write a system in the matrix form Y'=AY+G

1. What is the meaning of Y' in the system Y'=AY+G?

The Y' term represents the derivative of the vector Y, which is a function of time. It shows the rate of change of Y over time.

2. How is the matrix A related to the system Y'=AY+G?

The matrix A represents the coefficients of the variables in the system Y'=AY+G. It determines the behavior and stability of the system.

3. What does the matrix form of Y'=AY+G represent?

The matrix form of Y'=AY+G represents a linear system of differential equations. It is a way to express multiple equations in a compact form.

4. How do you solve a system in the matrix form Y'=AY+G?

The system can be solved by finding the eigenvalues and eigenvectors of the matrix A. These values can then be used to find the general solution of the system.

5. Can the system Y'=AY+G have multiple solutions?

Yes, the system can have multiple solutions depending on the initial conditions and the values of the matrix A. The solution can also change over time as the system evolves.

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