# Math Help - u and v solution

1. ## u and v solution

Show that if u and v are solutions of y'' + py = 0, then z= uv is a solution of (z''+ pz)' +pz' =0.

What are that saying???? Anything infor will help.Thanks in advance

2. Originally Posted by onemore
Show that if u and v are solutions of y'' + py = 0, then z= uv is a solution of (z''+ pz)' +pz' =0.

What are that saying???? Anything infor will help.Thanks in advance
I would think that what it is saying is clear- except whether p is a constant or a function of y. I would assume a constant. To show that z= uv is a solution to that, just go ahead and differentiate z= uv so that z'= u'v+ uv' by the product rule. Then z''= u''v+ 2u'v'+ uv''. (z"+ pz)'= (u''v+ 2u'v'+ uv'')'+ p(u'v+ uv')= u'''v+ u''v'+ 2u''v'+ 2u'v''+ u'v''+ uv'''+ pu'v+ puv'. Now notice that
u''v'+ puv'= (u''+ pu)v' and, since u satisfies y''+ py= 0, u''+ pu= 0. Similarly, u'v"+ pu'v= (v"+ pv)u'= 0.