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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

determine the general solution to the system x′= Ax for the given matrix A

Expert Solution

Step 1

Given: A system $x' = A x$ for the given matrix $A = [\begin{matrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \\ 4 & 5 & 6 & 7 \\ 7 & 6 & 5 & 4 \end{matrix}]$ .

To Find: The general solution of the system.

Step 2

A system $x' = A x$ for the given matrix $A = [\begin{matrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \\ 4 & 5 & 6 & 7 \\ 7 & 6 & 5 & 4 \end{matrix}]$ .

We have to find the characteristic equation as follows:

$\begin{array}{rcl} |A - λ I| & = & |\begin{matrix} 1 - λ & 2 & 3 & 4 \\ 4 & 3 - λ & 2 & 1 \\ 4 & 5 & 6 - λ & 7 \\ 7 & 6 & 5 & 4 - λ \end{matrix}| \\ = & λ^{2} (λ - 16) (λ + 2) \end{array}$

Now we have to solve the characteristic solution:

$λ^{2} (λ - 16) (λ + 2) = 0 λ = 0, λ = 0, λ = 16, λ = - 2$

Now we have to find the eigenvector corresponding to eigen value $λ = 0$ .

$\begin{array}{rcl} [A - 0 I] v_{1} & = & [\begin{matrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \\ 4 & 5 & 6 & 7 \\ 7 & 6 & 5 & 4 \end{matrix}] [\begin{matrix} a \\ b \\ c \\ d \end{matrix}] = 0 \\ a + 2 b + 3 c + 4 d & = & 0 \\ 4 a + 3 b + 2 c + d & = & 0 \\ 4 a + 5 b + 6 c + 7 d & = & 0 \\ 7 a + 6 b + 5 c + 4 d & = & 0 \end{array}$

Eigenvector is $[\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}]$ for $λ_{1} = λ_{2} = 0$ .

Now we have to find the eigenvector corresponding to the eigenvalue $λ_{3} = 16$ .

$\begin{array}{rcl} [A - 16 I] v_{3} & = & [\begin{matrix} - 15 & 2 & 3 & 4 \\ 4 & - 13 & 2 & 1 \\ 4 & 5 & - 10 & 7 \\ 7 & 6 & 5 & - 12 \end{matrix}] [\begin{matrix} a \\ b \\ c \\ d \end{matrix}] = 0 \\ - 15 a + 2 b + 3 c + 4 d & = & 0 \\ 4 a - 13 b + 2 c + d & = & 0 \\ 4 a + 5 b + - 10 c + 7 d & = & 0 \\ 7 a + 6 b + 5 c - 12 d & = & 0 \end{array}$

Eigenvector is $[\begin{matrix} 17 / 31 \\ 13 / 31 \\ 35 / 31 \\ 1 \end{matrix}]$ for $λ_{3} = 16$ .

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