Write the given system in the form x' = P(t)x+ f(t). x' = 3x-7y+z+t y' = x-5z+t²2² z' = 3y - 5z+t³ ---
Write the given system in the form x' = P(t)x+ f(t). x' = 3x-7y+z+t y' = x-5z+t²2² z' = 3y - 5z+t³ ---
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Converting a System of Differential Equations into Matrix Form**
Consider the system of differential equations given below:
\[
\begin{aligned}
x' &= 3x - 7y + z + t \\
y' &= x - 5z + t^2 \\
z' &= 3y - 5z + t^3
\end{aligned}
\]
We aim to write this system in the form:
\[
\mathbf{x}' = P(t)\mathbf{x} + f(t)
\]
Here’s the original differential system:
- \( x' = 3x - 7y + z + t \)
- \( y' = x - 5z + t^2 \)
- \( z' = 3y - 5z + t^3 \)
We can express the system in matrix notation by defining the vector \(\mathbf{x}\) and the matrix \(P(t)\) and vector \(f(t)\).
\[
\mathbf{x} = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
\]
Next, we identify the matrix \(P(t)\) from the coefficients of \(x\), \(y\), and \(z\):
\[
P(t) = \begin{pmatrix}
3 & -7 & 1 \\
1 & 0 & -5 \\
0 & 3 & -5
\end{pmatrix}
\]
We also identify the vector \(f(t)\) that consists of the non-homogeneous terms involving \(t\):
\[
f(t) = \begin{pmatrix}
t \\
t^2 \\
t^3
\end{pmatrix}
\]
So, the original system can be rewritten in matrix form as:
\[
\begin{pmatrix}
x' \\
y' \\
z'
\end{pmatrix}
=
\begin{pmatrix}
3 & -7 & 1 \\
1 & 0 & -5 \\
0 & 3 & -5
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
+
\begin{pmatrix}
t \\
t^2 \\
t^3
\end{pmatrix}
\]
Thus, the required matrix form is:
\[
\mathbf{x}' = P](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06bbd155-c9d1-4bdc-8881-0cd893ac016a%2Fce0c5544-1f62-49f6-b633-2a4e4f8d7450%2Fsj4ujx_processed.png&w=3840&q=75)
Transcribed Image Text:**Converting a System of Differential Equations into Matrix Form**
Consider the system of differential equations given below:
\[
\begin{aligned}
x' &= 3x - 7y + z + t \\
y' &= x - 5z + t^2 \\
z' &= 3y - 5z + t^3
\end{aligned}
\]
We aim to write this system in the form:
\[
\mathbf{x}' = P(t)\mathbf{x} + f(t)
\]
Here’s the original differential system:
- \( x' = 3x - 7y + z + t \)
- \( y' = x - 5z + t^2 \)
- \( z' = 3y - 5z + t^3 \)
We can express the system in matrix notation by defining the vector \(\mathbf{x}\) and the matrix \(P(t)\) and vector \(f(t)\).
\[
\mathbf{x} = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
\]
Next, we identify the matrix \(P(t)\) from the coefficients of \(x\), \(y\), and \(z\):
\[
P(t) = \begin{pmatrix}
3 & -7 & 1 \\
1 & 0 & -5 \\
0 & 3 & -5
\end{pmatrix}
\]
We also identify the vector \(f(t)\) that consists of the non-homogeneous terms involving \(t\):
\[
f(t) = \begin{pmatrix}
t \\
t^2 \\
t^3
\end{pmatrix}
\]
So, the original system can be rewritten in matrix form as:
\[
\begin{pmatrix}
x' \\
y' \\
z'
\end{pmatrix}
=
\begin{pmatrix}
3 & -7 & 1 \\
1 & 0 & -5 \\
0 & 3 & -5
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
+
\begin{pmatrix}
t \\
t^2 \\
t^3
\end{pmatrix}
\]
Thus, the required matrix form is:
\[
\mathbf{x}' = P
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