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4: Linear Systems of Ordinary Differential Equations (LSODE)

This page is a draft and is under active development. 

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Definition of Linear Systems of Ordinary Differential Equations (LSODE)

y1(t)=α11(t)y1(t) +α12(t)y2++α1n(t)yn+f1(t)y2(t)=α21(t)y1(t) +α22(t)y2++α2n(t)yn+f2(t)yn(t)=αn1(t)y1(t) +αn2(t)y2++αnn(t)yn+fn(t),

where yi(t):=dyi(t)dt denotes the first derivative of functions yi(t), i=1,2,, n, with respect to t.

Matrix Form of LSODE

yn(t)|y2(t)1]=αn1(t)|α21.(t)α11.(t)αn2(t)α22.(t)α12.(t)... αnn(t)α2n.(t)α1n.(t)]yn(t)|y2(t)y1.(t)]+fn(t)|f2(t)f1.(t)],

or in matrix form

y(t)=A(t)y(t)+f(t),

where

y(t)=|y2(ty1(t)1,A(t)=|α21.(tα11.(t)α22.(tα12.(t)α2n.(tα1n.(t)1,f(t)=|f2(tf1(t)1yn(t)αn1(t)αn2(t)αnn(t)fn(t), (3), A(t)in (3) is called coefficient matrix of (2) and f(t)


This page titled 4: Linear Systems of Ordinary Differential Equations (LSODE) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by William F. Trench.

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