4: Linear Systems of Ordinary Differential Equations (LSODE)
This page is a draft and is under active development.
( \newcommand{\kernel}{\mathrm{null}\,}\)
Definition of Linear Systems of Ordinary Differential Equations (LSODE)
y′1(t)=α11(t)y1(t) +α12(t)y2+…+α1n(t)yn+f1(t)y′2(t)=α21(t)y1(t) +α22(t)y2+…+α2n(t)yn+f2(t)y′n(t)=αn1(t)y1(t) +αn2(t)y2+…+αnn(t)yn+fn(t),
where y′i(t):=dyi(t)dt denotes the first derivative of functions yi(t), i=1,2,…, n, with respect to t.
Matrix Form of LSODE
⌊y′n(t)|y′2(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)=|⌈y′2(ty′1(t)⌉1,A(t)=|⌈α21.(tα11.(t)α22.(tα12.(t)α2n.(tα1n.(t)⌉1,f→(t)=|⌈f2(tf1(t)⌉1⌊y′n(t)⌋⌊αn1(t)αn2(t)αnn(t)⌋⌊fn(t)⌋, (3), A(t)in (3) is called coefficient matrix of (2) and f→(t)