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Eigenvalue method

A vector times an exponential turns a constant-coefficient system into an eigenvector problem; complex pairs then yield real solutions built from sines and cosines.

Notes on Diffy Qs: Differential Equations for Engineers, Eigenvalue method

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4 exercises

Exercise 101

Compute eigenvalues and eigenvectors of $A= \left[ \begin{smallmatrix} 1 & 0 & 3 \\ -1 & 0 & 1 \\ 2 & 0 & 2 \end{smallmatrix}\right]$ .
Solve the system $\vec{x}\,' = A \vec{x}$ .

Exercise 102

Compute eigenvalues and eigenvectors of $A=\left[ \begin{smallmatrix} 1 & 1 \\ -1 & 0 \end{smallmatrix}\right]$ .
Solve the system $\vec{x}\,' = A\vec{x}$ .

Exercise 103

Solve $x_1' = x_2$ , $x_2' = x_1$ using the eigenvalue method.

Exercise 104

Solve $x_1' = x_2$ , $x_2' = -x_1$ using the eigenvalue method.

Adapted from Notes on Diffy Qs © 2008–2026 Jiří Lebl. Licensed under CC BY-SA 4.0. Source: Notes on Diffy Qs: Differential Equations for Engineers. Imported at revision 658bcae.