Additional Exercises for Chapter 4

Second-Order Linear Systems

1. Write the second-order differential equation $$ \ddot x + 4 \dot x + 5 x = 0$$ as two first-order differential equations.

2. Write the linear system $$\dot x = 2x -3y, \ \dot y = 6x - y$$ as a second-order differential equation for \(x\), and then as a second-order equation for \(y\).

3. Without eliminating either of the variables, find the general solution of the system $$\dot x = x + 3 y, \ \dot y = 3x -7y. $$ Then find the specific solution that obeys the initial conditions \(x(0)=10, \ y(0)=0\).

4. For each of the following matrices, categorise the corresponding linear system by finding the trace and determinant.
   1. $$M=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$
   2. $$M=\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$$
   3. $$M=\begin{pmatrix} 2 & 2 \\ -3 & -2 \end{pmatrix}$$
   4. $$M=\begin{pmatrix} 2 & 2 \\ -2 & -1 \end{pmatrix}$$
5. Find the eigenvalues, and if necessary, the eigenvectors, of these matrices, and describe the behaviour of the linear system for large \( t\).
   1. $$ M = \begin{pmatrix} 2 & -3 \\ 4 & -4\end{pmatrix} $$
   2. $$ M = \begin{pmatrix} -1 & 1 \\ 6 & -2\end{pmatrix} $$
   3. $$ M = \begin{pmatrix} -1 & 2 \\ 2 & -4\end{pmatrix} $$
6. Describe the qualitative behaviour of the third-order system $$\dot x = x + y - 4 z, \ \dot y = - y, \ \dot z = 2x+3y -3z . $$

7. A matrix \(M\) is said to be idempotent if \(M^2=M\). Find an expression for the exponential of an idempotent matrix in terms of \(M, \ I\) and \( e \). Hence find the exponential of the matrix $$M=\begin{pmatrix} 1 & 3 \\ 0 & 0\end{pmatrix}.$$

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