Second-Order Linear ODEs
General Form
Characteristic Equation
Three Cases
Case 1: Distinct Real Roots ()
Case 2: Repeated Real Roots ()
Case 3: Complex Conjugate Roots ()
where ,
Examples
Example 1:
Characteristic equation:
Solution:
Example 2:
Characteristic equation:
Solution:
Example 3:
Characteristic equation:
Solution:
Linear Independence
Wronskian
Initial Value Problem
Given with , , substitute into the general solution and solve for .
Applications
Boundary Value Problems
Unlike IVPs, BVPs may have no solution, a unique solution, or infinitely many solutions.