Eigenvalues And Eigenvectors in DE

Description

While the desolve_system command can fully solve a given linear system of differential equations, it is still useful to know how to compute a general solution by finding the eigenvalues and eigenvectors of the corresponding matrix. Consider the system

(1)
\begin{align} x' &= -4x - 6y\\ y' &= x + 3y \end{align}

The corresponding matrix to this system is

(2)
\begin{align} A = \begin{bmatrix} -4 & -6 \\ 1 & 3 \end{bmatrix} \end{align}

We may use Sage to find both eigenvalues and an eigenvector for each eigenvalue:

Sage Cell

Notice that the .eigenvectors_right() method gives us the eigenvalue corresponding to each eigenvector, so the .eigenvalues() method is redundant in this case, bit it's still useful to know. From the output, we see we have two eigenvalues: $\lambda_1 = -3$ with eigenvector $\left<1, -1/6\right>$ and $\lambda_2 = 2$, with eigenvector $\left<1, -1\right>$. So, letting $c_1$ and $c_2$ be our arbitrary constants, the general solution to our system is

(3)
\begin{align} x &= c_1e^{-3t} + c_2e^{2t}\\ y &= -\frac{1}{6}c_1e^{-3t} - c_2e^{2t} \end{align}

Code

A = matrix(QQ, [[-4, -6], [1, 3]])
print(A.eigenvalues())
print(A.eigenvectors_right())

Options

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Date: 13 May 2020 04:31

Submitted by: Zane Corbiere

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