Reduction of Order
Repeated Roots of the Characteristic Equation
So far…
•
We’ve Learned How To Solve Second Order Linear
Homogeneous ODEs with Constant Coefficients
•
Characteristic Equation
•
We’ve seen distinct real roots, complex
conjugate roots
Characteristic Equation
Gives the Characteristic Equation
Case 1:
If roots are real and distinct,
and
General solution:
Case 2:
If roots are complex,
General solution:
and
Characteristic Equation
Gives the Characteristic Equation
What if there’s only one
root,
and
?
Characteristic Equation
Gives the Characteristic Equation
What if there’s only one
root,
Then we know one
solution:
How do we find another?
Reduction of Order
If we have a linear homogeneous second order equation
and we know one solution
How do we find another?
Reduction of Order
One solution
In general, finding solutions is difficult
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Can be solved with a first order linear ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Make a clever “renaming”
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
Make a clever “renaming”
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
First Order Linear
ODE
Reduction of Order
One solution
Guess second solution has the form:
Insert guess into ODE
First Order Linear
ODE
Second solution is
Use Wronskian to establish a Fundamental Set of
Solutions
Important Example
Characteristic Equation:
One
Solution
Important Example
One
Solution
Important Example
One
Solution
Important Example
One
Solution
Important Example
One
Solution
Important Example
One
Solution
Second Solution
Important Example
One
Solution:
Second Solution:
Wronskian
:
Wronskian
: Form a Fundamental Set of
Solutions
General
Solution:
This Holds in General
Gives the Characteristic Equation
Only one root,
and
This Holds in General
One solution:
Determine Reduction of Order Equation
This Holds in General
One solution:
Determine Reduction of Order Equation
This Holds in General
One solution:
Determine Reduction of Order Equation
This Holds in General
One solution:
Determine Reduction of Order Equation
This Holds in General
One solution:
Determine Reduction of Order Equation
Second Solution:
This Holds in General
If Characteristic Equation
Has only one root,
General Solution takes the
form:
Summary
•
We now know how to solve Second Order Linear
Homogeneous with Constant Coefficients:
•
Characteristic Equation
•
•
Distinct Real Roots, Complex Conjugate Roots,
One (Repeated) Root
If we have a Second Order Linear Homogeneous
Equation and one solution, use Reduction of Order
to find second solution.
Questions?