Solutions of Second
Order Linear ODEs
The Wronskian
Midterm Details
•
Midterm on WEDNESDAY, in class
•
Covers all sections up to 3.2 EXCEPT 2.4
•
Bring a Blue Book
•
1 Page (front and back) of Handwritten notes
•
50 minutes, 5 questions, 50 points
Solutions of Linear ODEs
Remember from Friday:
Solutions of Linear ODEs
Remember from Friday:
Has general solutions:
and
Solutions of Linear ODEs
Remember from Friday:
Has general solutions:
and
Which leads to the general form of solution:
Solutions of Linear ODEs
Remember from Friday:
Has general solutions:
and
Which leads to the general form of solution:
Or:
Solutions of Linear ODEs
Remember from Friday:
Has general solutions:
and
Which leads to the general form of solution:
Or:
Solutions of Linear ODEs
Remember from Friday:
Has general solutions:
and
Which leads to the general form of solution:
Or:
Solutions of Linear ODEs
Remember from Friday:
Has general solutions:
and
Which leads to the general form of solution:
Or:
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
Why
?
So
…
Solutions of Linear ODEs
This holds for general linear ODEs
If
and
Then so
are solutions,
is
In fact, so is
But that’s good!
Two constants allow us to fit initial conditions
By solving the equations
But can we always find
and
In other words, can we always
solve the system of equations?
?
The Wronskian
Can we always solve for
and
?
Recall the matrix formulation:
And the determinant
If the determinant of a matrix is non-zero,
there is a unique solution to the system of equations
The Wronskian
Can we always solve for
and
Take the determinant
If the determinant is NOT zero
Can always find
and
?
The Wronskian
What about different starting times?
Take the determinant
If the determinant is NOT zero
Can always find
and
The Wronskian
What about different starting times?
Take the determinant
If the determinant is NOT zero
Can always find
and
The Wronskian
What about different starting times?
Take the determinant
If the determinant is NOT zero
Can always find
and
The Wronskian
In general:
Take the determinant
If the determinant is NOT zero
Can always find
and
Is called the Wronskian function.
Fundamental Set
of Solutions
For two solutions of an
ODE
If the Wronskian is nonzero
The solutions are said to form a
Fundamental Set of Solutions
Because we can fit any initial conditions
Fundamental Set
of Solutions
To
verify
Form a fundamental set of solutions to
Verify
satisfy the
and
that
ODE
Compute the Wronskian
Verify the Wronskian is not
0
Example
Show
:
Form a fundamental set of solutions to
Verify
solutions:
Example
Show
:
Form a fundamental set of solutions to
(Repeat for
)
Compute Wronskian:
Example
Show
:
Form a fundamental set of solutions to
(Repeat for
)
Compute Wronskian:
Verify the Wronskian is not
0
But what if the Wronskian can become 0?
Abel’s Theorem
But what if the Wronskian can become 0?
Abel’s (Amazing) Theorem:
For linear homogeneous second order ODES:
The Wronskian for a set of solutions is
either
ALWAYS
ZERO
or
NEVER
ZERO
in the interval where the solution is exists.
Interval Where
Solution Exists
So how do we know interval where solution
exists?
Rewrite
as:
Interval Where
Solution Exists
So how do we know interval where solution
exists?
Rewrite
as:
Interval Where
Solution Exists
For
:
With initial conditions
A unique, second differentiable solution exists
on any open interval that contains
AND
where
and
and
are continuous.
Example
What is the longest interval where
has a unique solution?
Example
What is the longest interval where
has a unique solution?
Rewrite
as:
No
discontinuity
Discontinuity
at x=3
Discontinuity at
x=0 and x=3
Example
What is the longest interval where
has a unique solution?
Rewrite
as:
Discontinuity at
x=0 and x=3
Must contain
x=1
Example
What is the longest interval where
has a unique solution?
Rewrite
as:
Longest interval is
or
Summary
•
Linear combinations of solutions to homogeneous
linear ordinary differential equations are also
solutions.
•
The Wronskian determines where solutions form
fundamental sets of equations.
•
Existence and uniqueness are established by where
the coefficient functions are continuous.
Questions?