Prove the following true or false 1 Let f g and h be solutio

Prove the following true or false: 1) Let f, g, and h be solutions of y\'\'\'+p1(t)y\'\'+p2(t)y\'+p3(t)y=0 where p1(t), p2(t), and p3(t) are continuous on an open interval I. If they are linearly independent on I, then W(f, g, h) is never zero on I.

Solution

yes it is a true statement.

because if f , g and h are solutions of 3 rd ODE

if f , g and h are linearly independent then there exits 3 constants l , k and m such that l.f + k.g + m.h = 0 implies k , l,m=0

and the second order derivaties are continuous .

to satisfy above conditions the wronskin never zero