N-th Order Linear Differential Equations 179EXISTENCE OF N LINEARLY INDEPENDENT SOLUTIONS
TON-TH ORDER HOMOGENEOUS LINEAR DIFFERENTIA L
EQUATIONS
THEOREM 4.4 Let an(x), an-I(x), ... , aI(x), ao(x) be continuous on
the interval I and let an(x) ::j=. 0 for all x E I. There exist n linearly in-
dependent solutions on I of the n-th order homogeneous li near differential
equation
P roof: Let x 0 EI. By the existence and uniqueness theorem, there exist n
unique solutions on I, say YI (x ), y2(x ), ... , Yn(x), of then initial value prob-
lems consisting of the DE (19) and the following n sets of initial conditions
(20.1) YI(xo) = 1, YiI)(xo) = 0, Yi^2 >(xo) = 0 , Yin-I)(xo) = 0
(20.2) Y2(xo) = 0, ... ) y~n-I) (xo) =^0
(20.n) Yn(xo) = 0, y~I)(xo) = 0, y~^2 >(xo) = 0, ... , y~n-I)(xo) = 1
To prove that then solutions YI (x), y2(x), ... , Yn(x) of the specified n init ial
value problems are linearly independent solutions of the DE (19) on the
interval I, we examine their Wronskian evaluated at xo and find
YI (xo)
y~ (xo)Yn(xo)
y~(xo)(n-I)( ) (n-I)( ) (n-I)( )
YI Xo Y2 Xo · · · Yn-I Xo1 0 ... 0
0 1 ... 0= 1 ::j=.0.
0 0 ... 1
Hence, by Theorem 4.3 we know that the functions YI(x),y2(x),.. .,yn(x)
are li nearly independent solutions of the DE (19) on the interval I.