Ordinary Differential Equations

In the previous section, we shows that nontrivial solutions to (4.1.1) are of the form \(\vec{x}_i(t)=\vec{v}_ie^{\lambda_i t}\text{,}\) where \(\{\lambda_i,\vec{v}_i\}\) is an eigenpair of \(A\text{.}\)

Definition 4.2.1. Stability.

The origin \((0,0)\) is a critical point of the first-order autonomous system \(\vec{x}'=A\vec{x}\text{.}\) The origin can be classified as

• asymptotically stable if \(\mathrm{Re}(\lambda)<0\) for all eigenvalues \(\lambda\) of \(A\text{,}\)
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• stable if \(\mathrm{Re}(\lambda)=0\) for all eigenvalues \(\lambda\) of \(A\text{,}\) and
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• unstable if there exists an eigenvalue \(\lambda\) of \(A\) where \(\mathrm{Re}(\lambda)>0\text{.}\)
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Example 4.2.2. Distinct real eigenvalues.

Solve \(\vec{x}'=\begin{bmatrix}3& -2 \\ 2 & -2 \end{bmatrix}\vec{x}\text{.}\)

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Example 4.2.3. Distinct real eigenvalues.

Solve \(\vec{x}'=\begin{bmatrix}-2& 1 \\ 1 & -2 \end{bmatrix}\vec{x}, \vec{x}(0)=\begin{bmatrix} 2\\3 \end{bmatrix}\text{.}\)

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For a \(2\times 2\) matrix \(A\) with the eigenvalue \(\lambda\) repeated, there are two possibilities. Either (proper case) there are two independent eigenvectors associated to \(\lambda\text{,}\) i.e. the vector equation \((A-\lambda I)\vec{v}=\vec{0}\) has two free variables, or (degenerate case) there is only one eigenvector associated with \(\lambda\text{.}\)

Theorem 4.2.4. Solutions in the degenerate case.

Let \(\lambda\) be a real, repeated eigenvalue of \(A\) with \(\vec{v}\neq \vec{0}\) is its only corresponding eigenvector. Then the solutions to (4.1.1) are of the form

\begin{equation*} \vec{x}_1(t)=\vec{v}e^{\lambda t}, \quad \vec{x}_2(t)=(\vec{v}t+\vec{p})e^{\lambda t}, \end{equation*}

where \(\vec{p} \neq 0\) satisfies \((A-\lambda I)\vec{p}=\vec{v}\text{.}\)

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Proof.

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Definition 4.2.5. Generalized eigenvectors.

A nonzero vector \(\vec{p}\) is said to be a generalized eigenvector of \(A\) if \(\vec{p}\) satisfies \((A-\lambda I)\vec{p}=\vec{v}\text{.}\)

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Example 4.2.6. Repeated eigenvalues.

Solve \(\vec{x}'=\begin{bmatrix} 4& 0 \\ 0& 4\end{bmatrix}\vec{x}\text{.}\)

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Example 4.2.7. Repeated eigenvalues.

Solve \(\vec{x}'=\begin{bmatrix} 7& 1 \\ -4& 3\end{bmatrix}\vec{x}\text{.}\)

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Example 4.2.8. Repeated eigenvalues.

Solve \(\vec{x}'=\begin{bmatrix} 3& -4 \\ 1& -1\end{bmatrix}\vec{x}, \quad \vec{x}(0)=\begin{bmatrix} -1 \\ 3 \end{bmatrix}\text{.}\)

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Theorem 4.2.9. Complex eigenvalues for real-valued matrices.

Let \(A\) be an \(n \times n\) matrix with real-valued entries. If \(\lambda=\alpha+\beta i\) is an eigenvalue of \(A\text{,}\) then so is \(\overline{\lambda}=\alpha-\beta i\text{.}\) Furthermore, the corresponding eigenvectors to \(\lambda\) and \(\overline{\lambda}\) are complex conjugates.

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Proof.

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Example 4.2.10.

Find the eigenpairs of \(A=\begin{bmatrix} 0& 2 \\ -2& 0\end{bmatrix}\text{.}\)

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Remark 4.2.11. Useful tips when finding eigenvectors.

1. Always write complex eigenvalues in parentheses when calculating the correpsonding eigenvector. Otherwise, you might calculate the eigenvector associated to its complex conjugate.
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2. When checking if the rows are scalar multiples of one another, multiply the first row by the complex conjugate of the entry in the top right. Then check if the rows are scalar multiples of each other.
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3. Your eigenvector can look different depending on your choice of row to expand or choice of free variable. For our purposes, let’s expand by the top row.
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4. Since our eigenpairs are complex ocnjugates, we only need one eigenpair to find two real-valued, linearly independent solutions.
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Theorem 4.2.12. Solutions for complex eigenvalues.

Let \(\lambda=\alpha+\beta i\) be an eigenvalue of \(A\) with \(\vec{v}=\vec{a}+\vec{b}i\) as its corresponding eigenvector. Then the real-valued, linearly independent solutions to (4.1.1) are of the form

\begin{equation*} \vec{x}_1(t)=e^{\alpha t}\Big( \vec{a}\cos(\beta t) - \vec{b}\sin(\beta t)\Big), \end{equation*}

\begin{equation*} \vec{x}_2(t)=e^{\alpha t}\Big( \vec{a}\sin(\beta t)+\vec{b}\cos(\beta t) \Big). \end{equation*}

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Proof.

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Example 4.2.13. Complex eigenvalues.

Solve \(\vec{x}'=\begin{bmatrix} 0& 2 \\ -2& 0\end{bmatrix}\vec{x}, \quad \vec{x}(0)=\begin{bmatrix} 2 \\ -1 \end{bmatrix}\text{.}\)

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Example 4.2.14. Complex eigenvalues.

Solve \(\vec{x}'=\begin{bmatrix} 1& -4 \\ 2& -3\end{bmatrix}\vec{x}, \quad \vec{x}(0)=\begin{bmatrix} 6 \\ -2 \end{bmatrix}\text{.}\)

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Example 4.2.15. Complex eigenvalues.

Solve \(\vec{x}'=\begin{bmatrix} 2& -1 \\ 2& 0\end{bmatrix}\vec{x}, \quad \vec{x}(0)=\begin{bmatrix} 3 \\ -2 \end{bmatrix}\text{.}\)

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