Section 4.2  Linear Differential Operators

objective

• to recognize second order linear differential equations
• to understand how to use differential operators
• to be able to find solutions to second order linear differential equations given initial conditions

• coefficients: variable and constant

 
• homogeneous and nonhomogeneous equations

 
• operators: differential and linear

 
• linear combinations of solutions

 
• existence and uniqueness of solution

solved problems - using key ideas

• determining if an equation is linear or homogeneous with constant or variable coefficients, #1,3

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• understanding differential operators, #10

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• prove linearity of differential operators

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• Lemma 1, #11   (nonlinear operator)

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• prove Theorem 1 : linear combination of solutions

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• finding solutions to equations with initial conditions, 13a

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• D: differential operator notation, #23,33a,b

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• Theorem 2:  existence and uniqueness of solutions, #15,16, 21

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

QUESTION OF THE DAY

Use the Fundamental Theorem of Calculus to find the derivative of

SOLUTION