Homogeneous linear systems: constant coefficients: Difference between revisions
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r\vec{\xi}=A\vec{\xi} | r\vec{\xi}=A\vec{\xi} | ||
</math> | </math> | ||
Notice that \xi is an [[eigenvector]]. The entire eigenspace is solutions. | |||
Revision as of 15:11, 10 June 2024
Homogeneous linear systems of ODEs take the following form:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}'=Ax }
Similar to the scalar case, we look for some exponential solutions:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {x}}=e^{rt}{\vec {\xi }}}
Doing some basic differentiation and substitution from the above two equations, we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\vec{\xi}=A\vec{\xi} }
Notice that \xi is an eigenvector. The entire eigenspace is solutions.
