Lecture 07 - Jan 21

Ex x: Lec 7

Wronskian

if

Answer: Independence

Case 1: ,

Suppose

are L.I.

Case 2:

Consider

are L.I.

Remark: If are soln of , then is also a soln of .

If are soln of . Consider .

is a soln.

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If are real, then: are soln.

Suppose is complex. Then we have , .

The norm of the soln is defined by: .

The size of the absolute values is defined by:

Norm of a Solution

Let be a solution of . The norm of the soln is defined by .

Let be any soln of on an interval containing a point . Then for all :

Theorem

For , integrate from to :

By considering LHS of (2), we get

Then, we have

Suppose , then

From ① & ②, we get the requested result.

Existence theorem for IVP

For any real number, and constants , there exists a solution for the IVP,

Proof:

Let & be the soln of

We claim that there exists such that,

be a soln of the IVP,

is a soln of the IVP.

It has a unique soln,

Case 1:

Case 2: