Section 7.4

Note that phase portraits for Problems 1-16 are found in the answers for Section 7.2.

1. Saddle point (real eigenvalues of opposite sign)
2. Saddle point (real eigenvalues of opposite sign)
3. Saddle point (real eigenvalues of opposite sign)
4. Saddle point (real eigenvalues of opposite sign)
5. Saddle point (real eigenvalues of opposite sign)
6. Improper nodal source (distinct positive real eigenvalues)
7. Saddle point (real eigenvalues of opposite sign)
8. Center (pure imaginary eigenvalues)
9. Center (pure imaginary eigenvalues)
10. Center (pure imaginary eigenvalues)
11. Spiral source (complex conjugate eigenvalues with positive real part)
12. Spiral source (complex conjugate eigenvalues with positive real part)
13. Spiral source (complex conjugate eigenvalues with positive real part)
14. Spiral source (complex conjugate eigenvalues with positive real part)
15. Spiral source (complex conjugate eigenvalues with positive real part)
16. Improper nodal sink (distinct negative real eigenvalues)
17. Center; pure imaginary eigenvalues
18. Improper nodal source; distinct positive real eigenvalues; $v_{1} = {[\begin{array}{r} 0 & 1 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} - 1 & 1 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 0 & 1 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} - 1 & 1 \end{array}]}^{T}$
19. Saddle point; real eigenvalues of opposite sign; $v_{1} = {[\begin{array}{r} 0 & 1 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 0 & 1 \end{array}]}^{T}$ corresponds to the negativeeigen value and $v_{2} = {[\begin{array}{r} - 1 & 1 \end{array}]}^{T}$ $v_{2} = {[\begin{array}{r} - 1 & 1 \end{array}]}^{T}$ to the positive one.
20. Spiral source; complex conjugate eigenvalues with positive real part
21. Proper nodal source; repeated positive real eigenvalue with linearly independent eigenvectors
22. Parallel lines; one zero and one negative real eigenvalue
23. Spiral sink; complex conjugate eigenvalues with negative real part
24. Improper nodal sink; distinct negative real eigenvalues; $v_{1} = {[\begin{array}{r} 1 & 1 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} - 1 & 4 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 1 & 1 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} - 1 & 4 \end{array}]}^{T}$
25. Saddle point; real eigenvalues of opposite sign; $v_{1} = {[\begin{array}{r} 1 & 1 \end{array}]}^{T}$ $v_{1} = {[\begin{array}{r} 1 & 1 \end{array}]}^{T}$ corresponds to the positive eigenvalue and $v_{2} = {[\begin{array}{r} 4 & - 1 \end{array}]}^{T}$ $v_{2} = {[\begin{array}{r} 4 & - 1 \end{array}]}^{T}$ to the negative one.
26. Center; pure imaginary eigenvalues
27. Improper nodal source; distinct positive real eigenvalues; $v_{1} = {[\begin{array}{r} 2 & 3 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 2 & - 1 \end{array}]}^{T} .$ $v_{1} = {[\begin{array}{r} 2 & 3 \end{array}]}^{T}, v_{2} = {[\begin{array}{r} 2 & - 1 \end{array}]}^{T} .$
28. Spiral sink; complex conjugate eigenvalues with negative real part

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Table of Contents for Section 7.4

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Table of Contents for
Section 7.4