MATH 4B Lecture Notes - Lecture 8: Wronskian, Superposition Principle, Linear Algebra

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9 Feb 2017
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Second order linear homogenous ODE with constant coefficients:      
is a root of the characteristic equation
  
By substitution, we've seen that
  will be a solution
  : real distinct roots
  : complex conjugate roots
   

What is the other solution that is linearly independent from ?
  : repeated real roots
The nature of the roots depends on the discriminant   
Characteristic equation:  
 
Repeated real root   
 
Find the general solution of    
Solutions:
 
is called the variation of parameter
Substitute into    
  

   
  
 
Is there a function so that
 
?
For any constant ,
   is a solution
 

 
  
 
 
 
   
To check if
  and
  is a basis, compute the Wronskian
Example:
The general solution of    , when   is
  ,
  

Summary:
Repeated Roots
Second order linear homogeneous ODE: 
  
Then any linear combinations of and  are also solutions
Superposition principle: Suppose that  and are solutions of the second order
linear homogenous ODE
If  and are solutions whose Wronskian  , then
represent a
fundamental set of solutions
The general solution of the homogenous equation above is
 
for
arbitrary constants
Review of Homogeneous Case
Second order linear inhomogeneous ODE: 
  
Suppose that is some (particular) solution of the inhomogeneous equation
Suppose the  is the general solution of the homogeneous equation 
    
Inhomogeneous Case
The Superposition Principle for Inhomogeneous Equations
Lecture 8
Tuesday, February 7, 2017
5:34 PM
MATH 4B Page 1
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