Transformar ecuación diferencial en sistema de ecuaciones diferenciales

Converting Differential Equations into First Order Systems
An nth order differential equation can be converted into an n-dimensional system of first order
equations. There are various reasons for doing this, one being that a first order system is
much easier to solve numerically (using computer software) and most differential equations you
encounter in “real life” (physics, engineering etc) don’t have nice exact solutions.
If the equation is of order n and the unknown function is y, then set x1 = y, x2 = y0, . . . , xn =
y(n−1). Note (and then note again) that we only go up to the (n−1)st derivative in this process.
Examples:
(1) y(4) − 3y0y000 + sin(ty00) − 7ty2 = et. Set
x1 = y, x2 = y0, x3 = y00, x4 = y000
and then we have
x0
1 = y0 = x2
x0
2 = y00 = x3
x0
3 = y000 = x4
x0
4 = y(4) = 3y0y000 − sin(ty00) + 7ty2 + et = 3x2x4 − sin(tx3) + 7tx21
+ et
(2) y000 + 2y00 − y0 − 2y = 0. Set
x1 = y, x2 = y0, x3 = y00
and then we have
x0
1 = y0 = x2
x0
2 = y00 = x3
x0
3 = y000 = 2y + y0 − 2y00 = 2×1 + x2 − 2×3
Observe that the linear (homogeneous) equation is converted to a linear (homogeneous)
system x0 = Ax, where
A =
0
@
0 1 0
0 0 1
2 1 −2
1
A
Note further that
det(A − I) = −(3 + 22 −  − 2)
= −( − 1)( + 1)( + 2)
so the eigenvalues of A are  = 1,−1 and 2 which are the same as the roots of the
characteristic equation for the original 3rd order differential equation. This is always the
case for linear equations with constant coefficients.
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~ por leonsotelo en julio 7, 2014.