Chapter 10

Use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rr} 5 & 6 \\ 3 & -2 \end{array}\right) \mathbf{X} $$

In Problems \(1-8\), use the method of undetermined coefficients to solve the given system. \(\frac{d x}{d t}=2 x+3 y-7\) \(\frac{d y}{d t}=-x-2 y+5\)

Write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=3 x-5 y \\ &\frac{d y}{d t}=4 x+8 y \end{aligned} $$

The column vector \(\mathbf{X}=k\left(\begin{array}{l}4 \\\ 5\end{array}\right)\) is a solution of the linear system \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & 4 \\ 2 & -1\end{array}\right) \mathbf{X}-\left(\begin{array}{l}8 \\ 1\end{array}\right)\) for \(k=\) _________.

In Problems 1-12, find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=x+2 y \\ &\frac{d y}{d t}=4 x+3 y \end{aligned} $$

In Problems \(1-6\), write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=3 x-5 y \\ &\frac{d y}{d t}=4 x+8 y \end{aligned} $$

Use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{ll} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{array}\right) \mathbf{X} $$

Use the method of undetermined coefficients to solve the given system. \(\frac{d x}{d t}=5 x+9 y+2\) \(\frac{d y}{d t}=-x+11 y+6\)

Write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=4 x-7 y \\ &\frac{d y}{d t}=5 x \end{aligned} $$

In Problems, use \((3)\) to compute \(e^{A t}\) and \(e^{-\mathbf{A} t}\). $$ \mathbf{A}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) $$

Find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=2 x+2 y\\\ &\frac{d y}{d t}=x+3 y\\\ &\text { d } \end{aligned} $$

The vector \(\mathbf{X}=c_{1}\left(\begin{array}{r}-1 \\ 1\end{array}\right) e^{-9 t}+c_{2}\left(\begin{array}{l}5 \\ 3\end{array}\right) e^{7 t}\) is a solution of the initial-value problem \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & 10 \\ 6 & -3\end{array}\right) \mathbf{X}, \mathbf{X}(0)=\left(\begin{array}{l}2 \\\ 0\end{array}\right)\) for \(c_{1}= \ldots\) _________ and \(c_{2}=\) _________.

In Problems 1-8, use the method of undetermined coefficients to solve the given system. $$ \begin{aligned} &\frac{d x}{d t}=5 x+9 y+2 \\ &\frac{d y}{d t}=-x+11 y+6 \end{aligned} $$

In Problems 1-10, use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{ll} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{array}\right) \mathbf{X} $$

In Problems 1-12, find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=2 x+2 y \\ &\frac{d y}{d t}=x+3 y \end{aligned} $$

In Problems \(1-6\), write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=4 x-7 y \\ &\frac{d y}{d t}=5 x \end{aligned} $$

Use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{ll} 1 & \frac{1}{4} \\ 1 & 1 \end{array}\right) \mathbf{X} $$

Use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}1 & 3 \\ 3 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{l}-2 t^{2} \\ t+5\end{array}\right)\)

Write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=-3 x+4 y-9 z \\ &\frac{d y}{d t}=6 x-y \\ &\frac{d z}{d t}=10 x+4 y+3 z \end{aligned} $$

Find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=-4 x+2 y \\ &\frac{d y}{d t}=-\frac{5}{2} x+2 y \end{aligned} $$

In Problems 1-8, use the method of undetermined coefficients to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{ll} 1 & 3 \\ 3 & 1 \end{array}\right) \mathbf{X}+\left(\begin{array}{c} -2 t^{2} \\ t+5 \end{array}\right) $$

In Problems 1-10, use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{ll} 1 & \frac{1}{4} \\ 1 & 1 \end{array}\right) \mathbf{X} $$

In Problems 1-12, find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=-4 x+2 y \\ &\frac{d y}{d t}=-\frac{5}{2} x+2 y \end{aligned} $$

In Problems \(1-6\), write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=-3 x+4 y-9 z \\ &\frac{d y}{d t}=6 x-y \\ &\frac{d z}{d t}=10 x+4 y+3 z \end{aligned} $$

Use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right) \mathbf{X} $$

Write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=x-y \\ &\frac{d y}{d t}=x+2 z \\ &\frac{d z}{d t}=-x+z \end{aligned} $$

Find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=-\frac{5}{2} x+2 y \\ &\frac{d y}{d t}=\frac{3}{4} x-2 y \end{aligned} $$

Consider the linear system \(\mathbf{X}^{\prime}=\mathbf{A X}\) of two differential equations where \(\mathbf{A}\) is a real coefficient matrix. What is the general solution of the system if it is known that \(\lambda_{1}=1+2 i\) is an eigenvalue and \(\mathbf{K}_{1}=\left(\begin{array}{l}1 \\\ i\end{array}\right)\) is a corresponding eigenvector?

In Problems 1-8, use the method of undetermined coefficients to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rr} 1 & -4 \\ 4 & 1 \end{array}\right) \mathbf{X}+\left(\begin{array}{rr} 4 t+9 e^{6 t} \\ -t+e^{6 t} \end{array}\right) $$

In Problems 1-10, use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right) \mathbf{X} $$

In Problems 1-12, find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=-\frac{5}{2} x+2 y \\ &\frac{d y}{d t}=\frac{3}{4} x-2 y \end{aligned} $$

In Problems \(1-6\), write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=x-y \\ &\frac{d y}{d t}=x+2 z \\ &\frac{d z}{d t}=-x+z \end{aligned} $$

Use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} -1 & 3 & 0 \\ 3 & -1 & 0 \\ -2 & -2 & 6 \end{array}\right) \mathbf{x} $$

Use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}4 & \frac{1}{3} \\ 9 & 6\end{array}\right) \mathbf{X}+\left(\begin{array}{r}-3 \\\ 10\end{array}\right) e^{t}\)

Write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=x-y+z+t-1 \\ &\frac{d y}{d t}=2 x+y-z-3 t^{2} \\ &\frac{d z}{d t}=x+y+z+t^{2}-t+2 \end{aligned} $$

Find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rr} 10 & -5 \\ 8 & -12 \end{array}\right) \mathbf{X} $$

In Problems 1-8, use the method of undetermined coefficients to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{ll} 4 & \frac{1}{3} \\ 9 & 6 \end{array}\right) \mathbf{X}+\left(\begin{array}{r} -3 \\ 10 \end{array}\right) e^{t} $$

In Problems 1-10, use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} -1 & 3 & 0 \\ 3 & -1 & 0 \\ -2 & -2 & 6 \end{array}\right) \mathbf{X} $$

In Problems 1-12, find the general solution of the given system. $$ X^{\prime}=\left(\begin{array}{rr} 10 & -5 \\ 8 & -12 \end{array}\right) X $$

In Problems \(1-6\), write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=x-y+z+t-1 \\ &\frac{d y}{d t}=2 x+y-z-3 t^{2} \\ &\frac{d z}{d t}=x+y+z+t^{2}-t+2 \end{aligned} $$

Use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{lll} 1 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 1 \end{array}\right) \mathbf{X} $$

Use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}-1 & 5 \\ -1 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\sin t \\ -2 \cos t\end{array}\right)\)

Write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=-3 x+4 y+e^{-t} \sin 2 t \\ &\frac{d y}{d t}=5 x+9 z+4 e^{-t} \cos 2 t \\ &\frac{d z}{d t}=y+6 z-e^{-t} \end{aligned} $$

Find the general solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{ll} -6 & 2 \\ -3 & 1 \end{array}\right) \mathbf{X} $$

In Problems 5-14, solve the given linear system by the methods of this chapter. $$ \begin{aligned} &\frac{d x}{d t}=-4 x+2 y \\ &\frac{d y}{d t}=2 x-4 y \end{aligned} $$

In Problems 1-8, use the method of undetermined coefficients to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{cc} -1 & 5 \\ -1 & 1 \end{array}\right) \mathbf{X}+\left(\begin{array}{c} \sin t \\ -2 \cos t \end{array}\right) $$

In Problems 1-10, use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{lll} 1 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 1 \end{array}\right) \mathbf{X} $$

In Problems 1-12, find the general solution of the given system. $$ X^{\prime}=\left(\begin{array}{ll} -6 & 2 \\ -3 & 1 \end{array}\right) X $$

In Problems \(1-6\), write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=-3 x+4 y+e^{-t} \sin 2 t \\ &\frac{d y}{d t}=5 x+9 z+4 e^{-t} \cos 2 t \\ &\frac{d z}{d t}=y+6 z-e^{-t} \end{aligned} $$

Use diagonalization to solve the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{array}\right) \mathbf{X} $$