Chapter 10

In Problems 35-46, find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=2 x+y+2 z \\ &\frac{d y}{d t}=3 x+6 z \\ &\frac{d z}{d t}=-4 x-3 z \end{aligned} $$

Find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & -1 & 2 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right) \mathbf{x} $$

In Problems 35-46, find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & -1 & 2 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right) $$

Find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 4 & 0 & 1 \\ 0 & 6 & 0 \\ -4 & 0 & 4 \end{array}\right) \mathbf{X} $$

In Problems 35-46, find the general solution of the given system. $$ X^{\prime}=\left(\begin{array}{rrr} 4 & 0 & 1 \\ 0 & 6 & 0 \\ -4 & 0 & 4 \end{array}\right) X $$

Find the general solution of the given system. $$ \mathbf{x}^{\prime}=\left(\begin{array}{rrr} 2 & 5 & 1 \\ -5 & -6 & 4 \\ 0 & 0 & 2 \end{array}\right) \mathbf{X} $$

In Problems 35-46, find the general solution of the given system. $$ X^{\prime}=\left(\begin{array}{rrr} 2 & 5 & 1 \\ -5 & -6 & 4 \\ 0 & 0 & 2 \end{array}\right) X $$

Find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 2 & 4 & 4 \\ -1 & -2 & 0 \\ -1 & 0 & -2 \end{array}\right) \mathbf{x} $$

In Problems 35-46, find the general solution of the given system. $$ X^{\prime}=\left(\begin{array}{rrr} 2 & 4 & 4 \\ -1 & -2 & 0 \\ -1 & 0 & -2 \end{array}\right) X $$

Solve the given initial-value problem. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & -12 & -14 \\ 1 & 2 & -3 \\ 1 & 1 & -2 \end{array}\right) \mathbf{X}, \quad \mathbf{X}(0)=\left(\begin{array}{r} 4 \\ 6 \\ -7 \end{array}\right) $$

In Problems 47 and 48, solve the given initial-value problem. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & -12 & -14 \\ 1 & 2 & -3 \\ 1 & 1 & -2 \end{array}\right) \mathbf{X}, \quad \mathbf{X}(0)=\left(\begin{array}{r} 4 \\ 6 \\ -7 \end{array}\right) $$

Solve the given initial-value problem. $$ \mathbf{x}^{\prime}=\left(\begin{array}{rr} 6 & -1 \\ 5 & 4 \end{array}\right) \mathbf{x}, \quad \mathbf{x}(0)=\left(\begin{array}{r} -2 \\ 8 \end{array}\right) $$

In Problems 47 and 48, solve the given initial-value problem. $$ X^{\prime}=\left(\begin{array}{rr} 6 & -1 \\ 5 & 4 \end{array}\right) X, \quad X(0)=\left(\begin{array}{r} -2 \\ 8 \end{array}\right) $$

Solve each of the following linear systems. (a) \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right) \mathbf{X}\) (b) \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & 1 \\ -1 & -1\end{array}\right) \mathbf{X}\) Find a phase portrait of each system. What is the geometric significance of the line \(y=-x\) in each portrait?