Jan 28, Lect - 9
Existence Theorem
Let be constants and let be any real number. Then there exists a solution of on satisfying
Proof
Let be any linearly independent (LI) solutions of . We will show that there exist unique constants such that is a solution of , satisfying .
So,
The determinant of is the Wronskian and , since are LI solutions.
there exist unique constants such that,
is the solution of the IVP.
Uniqueness Theorem
Let be constants and let be any real number on any interval containing . Then there exists at most one solution of satisfying
Proof
Let and be any two solutions of the IVP .
Let .
So,
and,
where
Equations with Variable Coefficients
Consider,
then .
Theorem
If is the general solution of and is a particular solution of , then is the general solution of .
Proof
Given that is a solution of ,
is a solution of ,
Let .
Consider,
is a solution of .
Conversely,
Suppose is a solution of and is a particular solution of .
Then
is a solution of .
But we know that is a solution of . So,
Lemma
If are the solutions of
then the Wronskian is either zero or never zero.
Proof
Given that and are solutions of .
We have,
So,
.
Hence the lemma.
Reduction of Order
Theorem
If is a solution of
and for any .
Then, the transformation
reduces (1) to the first order equation,
where , and the second solution of (1) is given by,
Proof
Let be a solution of (1).
Suppose that is a solution of (1).
is a solution of (1) .
Note that from (2), if we set ,