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Exercise 8.8.3

Avg difficulty 3.0

Use matrix exponentials to find the solution to the first-order linear differential equation

[xy]′=[0−165][xy]\left[\begin{array}{c}x\\ y\end{array}\right]^{\prime}=\left[\begin{array}{rr}0&-1\\ 6&5\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]

with initial value

[x(0)y(0)]=[22].\left[\begin{array}{c}x(0)\\ y(0)\end{array}\right]=\left[\begin{array}{r}2\\ 2\end{array}\right].

Hint: form the matrix exponential eAte^{At} and then the solution is eAt v0e^{At}\,\mathbf{v}_{0}, where v0\mathbf{v}_{0} is the initial vector.

Selinger, Matrix Theory and Linear Algebra, Eigenvalues, eigenvectors, and diagonalization, Application: The matrix exponential, Exercise 8.8.3

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.