Download Linear differential equations with constant coefficients The general

Universitatea Babeş-Bolyai, Facultatea de Matematică şi Informatică
Secţia: Informatică engleză, Curs: Dynamical Systems, An: 2010/2011
Linear differential equations with constant coefficients
The general form of an nth order linear differential equation with constant coefficients is
(1)
x(n) + a1 x(n−1) + a2 x(n−2) + . . . an−1 x0 + an x = f (t),
where a1 , · · · , an ∈ R are called the coefficients and f ∈ C(I) (with I ⊂ R an open
interval) is called the nonhomogeneous term of (1). In the case that the function
f is not identically null, when discribing equation (1) we sometimes add the word
nonhomogeneous, while in the case that the function f is identically null equation
(1) becomes
(2)
x(n) + a1 x(n−1) + a2 x(n−2) + . . . an−1 x0 + an x = 0,
and, when discribing it, we add the word homogeneous.
The characteristic equation method is used to find a fundamental system of
solutions for (2). This method consists in the following steps.
Step 1. we write the characteristic equation associated to (2):
(3)
rn + a1 rn−1 + a2 rn−2 + · · · + an−1 r + an = 0.
Notice that this is an algebraic equation with the same coefficients as (2).
Step 2. we find all the n complex roots of (3).
Step 3. to the n roots of (3) we associate n functions following the rules:
r1 ∈ R simple 7−→ er1 t
r1 ∈ R of order µ 7−→ er1 t , ter1 t , . . . , tµ−1 er1 t
r1,2 = α ± i β (β 6= 0) simple 7−→ eα t cos β t, eα t sin β t
r1,2 = α ± i β (β 6= 0) of order µ 7−→ eα t cos β t, eα t sin β t, t eα t cos β t,
0
t eα t sin β t, . . . , tµ−1 eα t cos β t, tµ−1 eα t sin β t,
Theorem. The n functions found with the characteristic equation method form a
fundamental system of solutions for (2).
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Now we consider the linear nonhomogeneous equation (1) when the function f is
of some special form. We shall present the method of undetermined coefficients
used in order to find some particular solution of (1). This method is based on a simple
idea: when the function f is of some special form, the nonhomogeneous equation has
a particular solution of the same form. More precisely,
• when f (t) = A eα t and r = α is not a root of (3), then there exists a ∈ R such
that xp = a eα t is a particular solution of (1).
• when f (t) = (A0 + A1 t + · · · + Ak tk ) eα t and r = α is not a root of (3), then
there exists a0 , a1 , . . . , ak ∈ R such that xp = (a0 + a1 t + · · · + ak tk ) eα t is a
particular solution of (1).
• when f (t) = A eα t cos β t+B eα t sin β t and r = α+i β is not a root of (3), then
there exists a, b ∈ R such that xp = a eα t cos β t + b eα t sin β t is a particular
solution of (1).
• when the indicated numbers are simple roots (or double, ...) of (3), then the
given forms must be multiplied by t (or t2 , ...).
A very useful and simple result is the following. Before stating it, we denote by Lx
the left-hand side of equation (2).
The superposition principle. We consider f, g ∈ C(I), a, b ∈ R and the equation
(4)
Lx = a f (t) + b g(t).
Suppose that xp1 is some particular solution of Lx = f (t), and xp2 is some particular
solution of Lx = g(t).
Then xp = a xp1 + b xp2 is some particular solution of (4).
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