Solved: In 15 and 16 use a CAS or linear algebra software

In Problems 1–12 find the general solution of the given system. \(\frac{d x}{d t}=2 x+2 y\) \(\frac{d y}{d t}=x+3 y\) Text Transcription: dx/dt=2 x+2 y dy/dt=x+3 y

In Problems 1–12 find the general solution of the given system. \(\frac{d x}{d t}=-4 x+2 y\) \(\frac{d y}{d t}=-\frac{5}{2} x+2 y\) Text Transcription: dx/dt=-4 x+2 y dy/dt=-5/2 x+2 y

In Problems 1–12 find the general solution of the given system. \(\frac{d x}{d t}=x+2 y\) \(\frac{d y}{d t}=4 x+3 y\) Text Transcription: dx/dt=x+2 y dy/dt=4 x+3 y

In Problems 1–12 find the general solution of the given system. \(\frac{d x}{d t}=-\frac{5}{2} x+2 y\) \(\frac{d y}{d t}=\frac{3}{4} x-2 y\) Text Transcription: dx/dt=-5/2 x+2 y dy/dt=3/4 x-2 y

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

Examine your phase portraits in Problem 47. Under what conditions will the phase portrait of a 2 x 2 homogeneous linear system with complex eigenvalues consist of a family of closed curves? consist of a family of spirals? Under what conditions is the origin (0, 0) a repeller? An attractor?

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

In Problems 1–12 find the general solution of the given system. \(\frac{d x}{d t}=x+y-z\) \(\frac{d y}{d t}=2 y\) \(\frac{d z}{d t}=y-z\) Text Transcription: dx/dt=x+y-z dy/dt=2 y dz/dt=y-z

In Problems 1–12 find the general solution of the given system. \(\frac{d x}{d t}=2 x-7 y\) \(\frac{d y}{d t}=5 x+10 y+4 z\) \(\frac{d z}{d t}=5 y+2 z\) Text Transcription: dx/dt=2 x-7 y dy/dt=5 x+10 y+4 z dz/dt=5 y+2 z

In Problems 1–12 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} -1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 3 & -1 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr -1 & 1 & 0 1 & 2 & 1 0 & 3 & -1 {array}) mathbf X

In Problems 1–12 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} lll 1 & 0 & 1 0 & 1 & 0 1 & 0 & 1 {array}) mathbf X

In Problems 1–12 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} -1 & -1 & 0 \\ \frac{3}{4} & -\frac{3}{2} & 3 \\ \frac{1}{8} & \frac{1}{4} & -\frac{1}{2} \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr -1 & -1 & 0 3/4 & -3/2 & 3 1/8 & 1/4 & -1/2{array}) mathbf X

In Problems 1–12 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} -1 & 4 & 2 \\ 4 & -1 & -2 \\ 0 & 0 & 6 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr -1 & 4 & 2 4 & -1 & -2 0 & 0 & 6 {array}) mathbf X

In Problems 13 and 14 solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr} \frac{1}{2} & 0 \\ 1 & -\frac{1}{2} \end{array}\right) \mathbf{X} \); \(X(0)=\left(\begin{array}{l} 3 \\ 5 \end{array}\right) \) Text Transcription: mathbf X^prime=({array} rr 1/2 & 0 1 & -1/2{array}) mathbf X X(0)=({array} l 3 5 {array})

In Problems 13 and 14 solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{lll} 1 & 1 & 4 \\ 0 & 2 & 0 \\ 1 & 1 & 1 \end{array}\right) \mathbf{X} \); \(\mathbf{X}(0)=\left(\begin{array}{l} 1 \\ 3 \\ 0 \end{array}\right) \) Text Transcription: mathbf X^prime=({array} lll 1 & 1 & 4 0 & 2 & 0 1 & 1 & 1 {array}) mathbf X mathbf X 0=({array} l 1 3 0 end{array})

In Problems 15 and 16 use a CAS or linear algebra software as an aid in finding the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{lll} 0.9 & 2.1 & 3.2 \\ 0.7 & 6.5 & 4.2 \\ 1.1 & 1.7 & 3.4 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} lll 0.9 & 2.1 & 3.2 0.7 & 6.5 & 4.2 1.1 & 1.7 & 3.4 {array}) mathbf X

In Problems 15 and 16 use a CAS or linear algebra software as an aid in finding the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{clccc} 1 & 0 & 2 & -1.8 & 0 \\ 0 & 5.1 & 0 & -1 & 3 \\ 1 & 2 & -3 & 0 & 0 \\ 0 & 1 & -3.1 & 4 & 0 \\ -2.8 & 0 & 0 & 1.5 & 1 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} clccc 1 & 0 & 2 & -1.8 & 0 0 & 5.1 & 0 & -1 & 3 1 & 2 & -3 & 0 & 0 0 & 1 & -3.1 & 4 & 0 -2.8 & 0 & 0 & 1.5 & 1 {array}) mathbf X

Find phase portraits for the systems in Problems 2 and 4. For each system find any half-line trajectories and include these lines in your phase portrait.

In Problems 19–28 find the general solution of the given system. \(\frac{d x}{d t}=3 x-y\) \(\frac{d y}{d t}=9 x-3 y\) Text Transcription: dx/dt=3 x-y dy/dt=9 x-3 y

In Problems 19–28 find the general solution of the given system. \(\frac{d x}{d t}=-6 x+5 y\) \(\frac{d y}{d t}=-5 x+4 y\) Text Transcription: dx/dt=-6 x+5 y dy/dt=-5 x+4 y

In Problems 19–28 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll} -1 & 3 \\ -3 & 5 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} ll -1 & 3 -3 & 5 {array}) mathbf X

In Problems 19–28 find the general solution of the given system. \(\frac{d x}{d t}=3 x-y-z\) \(\frac{d y}{d t}=x+y-z\) \(\frac{d z}{d t}=x-y+z\) Text Transcription: dx/dt=3 x-y-z dy/dt=x+y-z dz/dt=x-y+z

In Problems 19–28 find the general solution of the given system. \(\frac{d x}{d t}=3 x+2 y+4 z\) \(\frac{d y}{d t}=2 x+2 z\) \(\frac{d z}{d t}=4 x+2 y+3 z\) Text Transcription: dx/dt=3 x+2 y+4 z dy/dt=2 x+2 z dz/dt=4 x+2 y+3 z

(a) Use computer software to obtain the phase portrait of the system in Problem 5. If possible, include arrowheads as in Figure 8.2.2. Also include four half-lines in your phase portrait. (b) Obtain the Cartesian equations of each of the four half-lines in part (a). (c) Draw the eigenvectors on your phase portrait of the system.

In Problems 19–28 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr} 12 & -9 \\ 4 & 0 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rr 12 & -9 4 & 0 {array}) mathbf X

In Problems 19–28 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & -1 & 1 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr 1 & 0 & 0 0 & 3 & 1 0 & -1 & 1 {array}) mathbf X

In Problems 19–28 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} 5 & -4 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & 5 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr 5 & -4 & 0 1 & 0 & 2 0 & 2 & 5 {array}) mathbf X

In Problems 19–28 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & 0 & 0 \\ 2 & 2 & -1 \\ 0 & 1 & 0 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr 1 & 0 & 0 2 & 2 & -1 0 & 1 & 0 {array}) mathbf X

Show that the 5 x 5 matrix \(\mathbf{A}=\left(\begin{array}{lllll} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 0 & 2 \end{array}\right) \) has an eigenvalue \(\lambda_{1}\) of multiplicity 5. Show that three linearly independent eigenvectors corresponding to \(\lambda_{1}\) can be found. Text Transcription: mathbf A =({array} lllll 2 & 1 & 0 & 0 & 0 0 & 2 & 0 & 0 & 0 0 & 0 & 2 & 0 & 0 0 & 0 & 0 & 2 & 1 0 & 0 & 0 & 0 & 2 {array}) lambda_1

In Problems 19–28 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{lll} 4 & 1 & 0 \\ 0 & 4 & 1 \\ 0 & 0 & 4 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} lll 4 & 1 & 0 0 & 4 & 1 0 & 0 & 4 {array}) mathbf X

In Problems 29 and 30 solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr} 2 & 4 \\ -1 & 6 \end{array}\right) \mathbf{X} \); \(\mathbf{X}(0)=\left(\begin{array}{r} -1 \\ 6 \end{array}\right) \) Text Transcription: mathbf X^prime=({array} rr 2 & 4 -1 & 6 {array}) mathbf X mathbf X 0=({array} r -1 6 {array})

In Problems 33–44 find the general solution of the given system. \(\frac{d x}{d t}=6 x-y\) \(\frac{d y}{d t}=5 x+2 y\) Text Transcription: dx/dt=6 x-y dy/dt=5 x+2 y

In Problems 29 and 30 solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right) \mathbf{X} \); \(\mathbf{X}(0)=\left(\begin{array}{l} 1 \\ 2 \\ 5 \end{array}\right) \) Text Transcription: mathbf X^prime=({array} lll 0 & 0 & 1 0 & 1 & 0 1 & 0 & 0 {array}) mathbf X mathbf X 0=({array} l 1 2 5 {array})

In Problems 33–44 find the general solution of the given system. \(\frac{d x}{d t}=x+y\) \(\frac{d y}{d t}=-2 x-y\) Text Transcription: dx/dt=x+y dy/dt=-2 x-y

In Problems 33–44 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll} 1 & -8 \\ 1 & -3 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} ll 1 & -8 1 & -3 {array}) mathbf X

In Problems 33–44 find the general solution of the given system. \(\frac{d x}{d t}=5 x+y\) \(\frac{d y}{d t}=-2 x+3 y\) Text Transcription: dx/dt=5 x+y dy/dt=-2 x+3 y

In Problems 33–44 find the general solution of the given system. \(\frac{d x}{d t}=4 x+5 y\) \(\frac{d y}{d t}=-2 x+6 y\) Text Transcription: dx/dt=4 x+5 y dy/dt=-2 x+6 y

In Problems 33–44 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll} 4 & -5 \\ 5 & -4 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} ll 4 & -5 5 & -4 {array}) mathbf X

In Problems 33–44 find the general solution of the given system. \(\frac{d x}{d t}=2 x+y+2 z\) \(\frac{d y}{d t}=3 x+6 z\) \(\frac{d z}{d t}=-4 x-3 z\) Text Transcription: dx/dt=2 x+y+2 z dy/dt=3 x+6 z dz/dt=-4 x-3 z

In Problems 33–44 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & -1 & 2 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right) \mathbf{X}\) Text Transcription: mathbf X^prime=({array} rrr 1 & -1 & 2 -1 & 1 & 0 -1 & 0 & 1 {array}) mathbf X

In Problems 33–44 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} 2 & 4 & 4 \\ -1 & -2 & 0 \\ -1 & 0 & -2 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr 2 & 4 & 4 -1 & -2 & 0 -1 & 0 & -2 {array}) mathbf X

In Problems 33–44 find the general solution of the given system. \(\frac{d x}{d t}=z\) \(\frac{d y}{d t}=-z\) \(\frac{d z}{d t}=y\) Text Transcription: dx/dt=z dy/dt=-z dz/dt=y

In Problems 33–44 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} 4 & 0 & 1 \\ 0 & 6 & 0 \\ -4 & 0 & 4 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr 4 & 0 & 1 0 & 6 & 0 -4 & 0 & 4 {array}) mathbf X

In Problems 33–44 find the general solution of the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} 2 & 5 & 1 \\ -5 & -6 & 4 \\ 0 & 0 & 2 \end{array}\right) \mathbf{X} \) Text Transcription: mathbf X^prime=({array} rrr 2 & 5 & 1 -5 & -6 & 4 0 & 0 & 2 {array}) mathbf X

In Problems 45 and 46 solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & -12 & -14 \\ 1 & 2 & -3 \\ 1 & 1 & -2 \end{array}\right) \mathbf{X} \), \(\mathbf{X}(0)=\left(\begin{array}{r} 4 \\ 6 \\ -7 \end{array}\right) \) Text Transcription: mathbf X^prime=({array} rrr 1 & -12 & -14 1 & 2 & -3 1 & 1 & -2 {array}) mathbf X mathbf X 0=({array} r 4 6 -7 {array})

Find phase portraits for the systems in Problems 36, 37, and 38.

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? Text Transcription: mathbf X^prime=({array} ll 1 & 1 1 & 1 {array}) mathbf X mathbf X^prime=({array} rr 1 & 1 -1 & -1 {array}) mathbf X

(a) Solve (2) of Section 7.6 using the first method outlined in the Remarks (page 345)—that is, express (2) of Section 7.6 as a first-order system of four linear equations. Use a CAS or linear algebra software as an aid in finding eigenvalues and eigenvectors of a 4 x 4 matrix. Then apply the initial conditions to your general solution to obtain (4) of Section 7.6. (b) Solve (2) of Section 7.6 using the second method out-lined in the Remarks—that is, express (2) of Section 7.6 as a second-order system of two linear equations. Assume solutions of the form \(\mathbf{X}=\mathbf{V} \sin \omega t\) and\(\mathbf{X}=\mathbf{V} \cos \omega t\). Find the eigenvalues and eigenvectors of a 2 x 2 matrix. As in part (a), obtain (4) of Section 7.6. Text Transcription: mathbf X=mathbf V sin omega t mathbf X=mathbf V cos omega t

Consider the 5 x 5 matrix given in Problem 31. Solve the system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}\) without the aid of matrix methods, but write the general solution using matrix notation. Use the general solution as a basis for a discussion of how the system can be solved using the matrix methods of this section. Carry out your ideas. Text Transcription: mathbf X^prime=mathbf A mathbf X

Obtain a Cartesian equation of the curve define parametrically by the solution of the linear system in Example 6. Identify the curve passing through (2, 1) in Figure 8.2.5. [Hint: Compute \(x^{2}, y^{2}\), and xy.] Text Transcription: x^2 y^2

In Problems 45 and 46 solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr} 6 & -1 \\ 5 & 4 \end{array}\right) \mathbf{X} \), \(\mathbf{X}(0)=\left(\begin{array}{r} -2 \\ 8 \end{array}\right) \) Text Transcription: mathbf X^prime=({array} rr 6 & -1 5 & 4 {array}) mathbf X mathbf X 0=({array} r -2 8 {array})

Find phase portraits for the systems in Problems 20 and 21. For each system find any half-line trajectories and include these lines in your phase portrait.