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Write the (third-order) differential equation y"" = 2y" +9y' + 6y as a system of (first-order) differential equations. Namely, determine a matrix A such that y' = Ay where y = 園 Submit

Write the (third-order) differential equation y"" = 2y" +9y' + 6y as a system of (first-order) differential equations. Namely, determine a matrix A such that y' = Ay where y = 園 Submit

Advanced Engineering Mathematics
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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**Problem Statement:** Write the (third-order) differential equation \( y''' = 2y'' + 9y' + 6y \) as a system of (first-order) differential equations. Namely, determine a matrix \( A \) such that \( \vec{y}' = A\vec{y} \) where \( \vec{y} = \begin{bmatrix} y \\ y' \\ y'' \end{bmatrix} \). **Matrix Representation:** The problem requires transforming a third-order differential equation into a system of first-order differential equations. Specifically, we need to express the third derivative \( y''' \) using a matrix equation. **Solution Steps:** 1. **Define \( \vec{y} \):** - \(\vec{y} = \begin{bmatrix} y \\ y' \\ y'' \end{bmatrix}\) 2. **Express \( \vec{y}' \):** - \(\vec{y}' = \begin{bmatrix} y' \\ y'' \\ y''' \end{bmatrix}\) 3. **Set Up the Matrix \( A \):** - Use the original third-order equation \( y''' = 2y'' + 9y' + 6y \) to express \( y''', y'', \text{and } y' \) in terms of matrix multiplication. - \(\vec{y}' = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 6 & 9 & 2 \end{bmatrix} \begin{bmatrix} y \\ y' \\ y'' \end{bmatrix}\) The empty grid under the statement represents where the matrix \( A \) is to be entered. The "Submit" button implies this is an interactive component on the website where users input their solution.
Transcribed Image Text:**Problem Statement:** Write the (third-order) differential equation \( y''' = 2y'' + 9y' + 6y \) as a system of (first-order) differential equations. Namely, determine a matrix \( A \) such that \( \vec{y}' = A\vec{y} \) where \( \vec{y} = \begin{bmatrix} y \\ y' \\ y'' \end{bmatrix} \). **Matrix Representation:** The problem requires transforming a third-order differential equation into a system of first-order differential equations. Specifically, we need to express the third derivative \( y''' \) using a matrix equation. **Solution Steps:** 1. **Define \( \vec{y} \):** - \(\vec{y} = \begin{bmatrix} y \\ y' \\ y'' \end{bmatrix}\) 2. **Express \( \vec{y}' \):** - \(\vec{y}' = \begin{bmatrix} y' \\ y'' \\ y''' \end{bmatrix}\) 3. **Set Up the Matrix \( A \):** - Use the original third-order equation \( y''' = 2y'' + 9y' + 6y \) to express \( y''', y'', \text{and } y' \) in terms of matrix multiplication. - \(\vec{y}' = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 6 & 9 & 2 \end{bmatrix} \begin{bmatrix} y \\ y' \\ y'' \end{bmatrix}\) The empty grid under the statement represents where the matrix \( A \) is to be entered. The "Submit" button implies this is an interactive component on the website where users input their solution.
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