There are no other solutions... how to prove it?
Bill Dubuque
>
> how can I prove that the ONLY solutions of the ODE
>
> y'' = -y are the ones given by A cos(x) + B sin(x) ?
Easy: Wronskian W(cos,sin) = cos^2 + sin^2 = 1 != 0, so
Variation of Parameters (2) specialized to _homogeneous_
case shows that any other solution is a linear combination
of cos, sin with _constant_ coef's. Below are easy proofs,
which generalizes to higher-order LODEs (and recurrences).
The proof is slicker in matrix form, e.g. Lemma 4.1 in (3).
THEOREM If f,g,h are solutions on an interval I of the LODE
y'' = p y' + q y, p,q continuous on I
and gh'-g'h != 0 for all x in I, then there exist constants c,d
such that f = c g + d h on I
PROOF: The below equations [0],[1] have a unique solution (c,d)
since det = W(g,h) = gh'- g'h != 0 on I.
[0] f = c g + d h
[1] f' = c g' + d h'
[2] qf+pf' = f" = c g" + d h" via q[0]+p[1], g"=qg+pg', h"=qh+ph'
[3] 0 = c'g + d'h via [0]'-[1]
[4] 0 = c'g' + d'h' via [1]'-[2] (1b)
The above equations [3],[4] have unique solution (c',d') = (0,0)
since det = gh'-g'h = W(g,h) != 0 on I. Thus c,d are constants.
-------
THEOREM If f,g,h are solutions of the recurrence
y'' = p y' + q y, where y'(n) := y(n+1)
with Wronskian W = gh'-g'h != 0 then there exist constants c,d
such that f = c g + d h
PROOF: [0],[1] below have unique solution (c,d) via det = W != 0
[0] f = c g + d h
[1] f' = c g' + d h' Now q[0] + p[1] yields:
[2] qf+pf' = f" = c g" + d h" via qg+pg' = g", qh+ph' = h"
[3] 0 = (c'-c)g' + (d'-d)h' via [0]'-[1]
[4] 0 = (c'-c)g" + (d'-d)h" via [1]'-[2]
The above equations [3],[4] have solution (c'-c,d'-d) = (0,0),
unique via det = W' = g'h"-g"h' != 0. So c,d are constants,
since c' = c means c(n+1) = c(n).
-Bill Dubuque
(1) L. E. Pursell: A simple uniqueness theory for ordinary linear
homogeneous differential equations, Amer. Math. Monthly, 74, 1967, 47-50
http://links.jstor.org/sici?sici=0002-9890(196701)74:1%3C47%3E
(2) Variation of Parameters
http://planetmath.org/encyclopedia/VariationOfParameters.html
http://ltcconline.net/greenl/courses/204/appsHigherOrder/variationHigher.htm
(3) Marius van der Put: Symbolic analysis of differential equations
http://msri.org/activities/programs/9899/focm/soggy/MSRIintro/van_der_Put2.ps
d:\Symbolic analysis of differential equations -- Put.ps.gz