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Inhomogeneous Linear Systems

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https://texercises.com/exercise/inhomogeneous-linear-systems/

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Exercise:
A system of inhomogeneous linear differential s is given by fractextrmdtextrmdtbf x bf A bf x+bf b with bf x pmatrix x_ x_ vdots x_n pmatrix and the constant vector bf b pmatrix b_ b_ vdots b_n pmatrix. Show that this is equivalent to a homogeneous system of differential s.

Solution:
The fixed po bf x^* of the system is given by bf Abf x^*+bf b bf Longrightarrow bf Abf x^* -bf b We define a new vector function bf ut as follows: bf ut bf xt-bf x^* The corresponding differential s are given by fractextrmdtextrmdtbf ut fractextrmdtextrmdtleftbf xt-bf x^*right fractextrmdtextrmdtbf xt bf A bf xt+bf b bf A leftbf ut + bf x^*right + bf b bf A bf ut + bf A bf x^* + bf b bf A bf ut - bf b + bf b bf A bf ut The system of differential s for bf ut is defined by the same matrix bf A but with no constant vector i.e. it is a homogeneous system. The fixed po is therefore the origin.

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Branches	Differential equations
Tags	fixed point, homogeneous, inhomogeneous, linear differential equations
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Difficulty	(2, default)
Points	0 (default)
Language	(English)
Type	Calculative / Quantity
Creator	by
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