91Ó°ÊÓ

Find the general solution of each of the homogeneous linear systems in Exercises \(1-24\), where in each exercise $$\mathbf{x}=\left(\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right)$$ $$ \frac{d \mathbf{x}}{d t}=\left(\begin{array}{rrr} 1 & 9 & 9 \\ 0 & 19 & 18 \\ 0 & 9 & 10 \end{array}\right) \mathbf{x} $$

Find the general solution of each of the linear systems in Exercises \(1-22\) $$ \begin{aligned} &\frac{d x}{d t}=4 x-2 y \\ &\frac{d y}{d t}=5 x+2 y \end{aligned} $$

Find the general solution of each of the homogeneous linear systems in Exercises \(1-18\), using the vector-matrix methods of this section, where in each exercise $$\mathbf{x}=\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right)$$ $$ \frac{d x}{d t}=\left(\begin{array}{rr} 5 & 4 \\ -1 & 1 \end{array}\right) x $$

Use the operator method described in this section to find the general solution of each of the linear systems in Exercises 1-26. $$ \begin{aligned} &2 x^{\prime}+y^{\prime}+x+5 y=4 t \\ &x^{\prime}+y^{\prime}+2 x+2 y=2 \end{aligned} $$

Find the general solution of each of the homogeneous linear systems in Exercises \(1-18\), using the vector-matrix methods of this section, where in each exercise $$\mathbf{x}=\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right)$$ $$ \frac{d x}{d t}=\left(\begin{array}{ll} 1 & -2 \\ 2 & -3 \end{array}\right) \mathbf{x} $$

Use the operator method described in this section to find the general solution of each of the linear systems in Exercises 1-26. $$ \begin{aligned} &2 x^{\prime}+y^{\prime}+x+y=t^{2}+4 t \\ &x^{\prime}+y^{\prime}+2 x+2 y=2 t^{2}-2 t \end{aligned} $$

Find the general solution of each of the linear systems in Exercises \(1-22\) $$ \begin{aligned} &\frac{d x}{d t}=x-5 y \\ &\frac{d y}{d t}=2 x-y \end{aligned} $$

Use the operator method described in this section to find the general solution of each of the following linear systems. $$ \begin{aligned} &2 \frac{d x}{d t}+\frac{d y}{d t}-x-y=1 \\ &\frac{d x}{d t}+\frac{d y}{d t}+2 x-y=t \end{aligned} $$

Find the general solution of each of the linear systems in Exercises \(1-22\) $$ \begin{aligned} &\frac{d x}{d t}=3 x-2 y \\ &\frac{d y}{d t}=2 x+3 y \end{aligned} $$

Find the general solution of each of the homogeneous linear systems in Exercises \(1-24\), where in each exercise $$\mathbf{x}=\left(\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right)$$ $$ \frac{d \mathbf{x}}{d t}=\left(\begin{array}{rrr} -5 & -3 & -3 \\ 8 & 5 & 7 \\ -2 & -1 & -3 \end{array}\right) \mathbf{x} $$