Section 4.3  Fundamental Solutions of Homogeneous Equations

objective

• to be able to determine whether functions are linearly dependent or linearly independent
• to be able to find general solutions when given solutions
• to be able to find particular solutions when given solutions and initial conditions

• general solution

 
• representation of solutions (homogeneous case)

 
• Wronskian, W[y₁,y₂](x)

 
• Fundamental solution set

 
• linearly independent vectors

 
• linearly dependent vectors

 
• linear dependence of functions

 
• Theorem 4 - condition for linear dependence of solutions

 
• Corollary 1 - the equivalence of the statements concerning fundamental sets, linear independence, and the Wronskian

solved problems - using key ideas

• determining linear dependency of functions using the definition, #1,5

 

 
 
 
 
 
 
 
 
 
 
 
 

• finding a general solution, #9

 

 
 
 
 
 
 
 
 
 
 

• examining solution sets of a given differential equation, #13

 

 
 
 
 
 
 
 
 
 
 
 
 

• studying Wronskians, #14a,b

 

 
 
 
 
 
 
 
 
 
 

• linear dependence of three functions, #24,25a

 

 
 
 
 
 
 
 
 
 
 
 

• Wronskian of three functions, #27a

 

 
 
 
 
 
 
 
 
 
 
 

QUESTION OF THE DAY

SOLUTION

SOLUTION.