Section 5.5. Solve a system of homogeneous differential equations using the coefficient matrix
Objective:
1. Solve a system of homogeneous differential equations using the coefficient matrix with complex eigenvalues
2. Understand connection between the phase plan and the solution
In previous section, we only work on cases that the eigenvalues of the coefficient matrix are real numbers. At here, we use the same principal to solve the case that the eigenvalues are complex numbers. We know that if a polynomial has real numbers as coefficients then its complex roots must be a pair, i.e. if [latex]a+bi[/latex] is a solution of [latex]f(x)=0[/latex] then [latex]a-bi[/latex] is also a solution of [latex]f(x)=0[/latex]. Finding the eigenvalues of a matrix [latex]A[/latex], we solve [latex]\text{det}(A-\lambda I)=0[/latex]. Since the entries of [latex]A[/latex] are all real numbers, [latex]\text{det}(A-\lambda I)=0[/latex] is a real polynomial. This means if [latex]a+bi[/latex] is an eigenvalue of [latex]A[/latex] then [latex]a-bi[/latex] is also an eigenvalue of [latex]A[/latex]. Not only that, if [latex]\overrightarrow{v}=\overrightarrow{p}+i\overrightarrow{q}[/latex] is an eigenvector of [latex]A[/latex] such that all entries of [latex]\overrightarrow{p}[/latex] and [latex]\overrightarrow{q}[/latex] are real, then [latex]\overrightarrow{w}=\overrightarrow{p}-i\overrightarrow{q}[/latex] is also an eigenvector of [latex]A[/latex].
Proof for the vector part:
Example 1: Given [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -1 & 1\\ -1 & -1 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}[/latex], find the general solution of the system of equations. Sketch its phase plane.
Exercise 1: Given [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -2 & 4\\ -1 & -2 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}[/latex], find the general solution of the system of equations. Sketch its phase plane.
Example 2: Given [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 2 & -4\\ 2 & -2 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}[/latex], find the general solution of the system of equations. Sketch its phase plane.
Exercise 2: Given [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -1 & 2\\ -5 & 1 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}[/latex], find the general solution of the system of equations. Sketch its phase plane.
Example 3: Given [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 2 & 2\\ -2 & 2 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}[/latex], find the general solution of the system of equations. Sketch its phase plane.
Exercise 3:Given [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 3 & 5\\ -5 & 3 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}[/latex], find the general solution of the system of equations. Sketch its phase plane.
Example 4: Given [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 1 & -5\\ 1 & -3 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}[/latex], find the general solution of the system of equations. Sketch its phase plane.
Exercise 4: Given [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -2 & 4\\ -4 & 4 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}[/latex], find the general solution of the system of equations. Sketch its phase plane.
Group Work:
1. Solve the initial value problem and describe the behavior of the
solution as [latex]t[/latex] approaches infinity.
(a) [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 1 & -5\\ 1 & -3 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right],[/latex] [latex]\left[\begin{array}{c} x_{1}(0)\\ x_{2}(0) \end{array}\right]=\left[\begin{array}{c} 1\\ 1 \end{array}\right][/latex].
(b) [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -3 & 2\\ -1 & -1 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right],[/latex] [latex]\left[\begin{array}{c} x_{1}(0)\\ x_{2}(0) \end{array}\right]=\left[\begin{array}{c} 1\\ -2 \end{array}\right][/latex].
(c) [latex]\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} \frac{3}{4} & -2\\ 1 & -\frac{5}{4} \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right],[/latex] [latex]\left[\begin{array}{c} x_{1}(0)\\ x_{2}(0) \end{array}\right]=\left[\begin{array}{c} 0\\ -1 \end{array}\right][/latex].