Mathematics - Differential Equations - Second-order ODEs - Second-Order Linear ODEs

12 steps: (1) Form: y''+p(x)y'+q(x)y = g(x). (2) Homogeneous (g=0): y''+ay'+by=0. (3) CHARACTERISTIC EQUATION: r²+ar+b=0. (4) Case 1: Two distinct real roots r₁,r₂. y=C₁e^(r₁x)+C₂e^(r₂x). (5) Example: y''-5y'+6y=0. r²-5r+6=0. r=2,3. y=C₁e²ˣ+C₂e³ˣ. (6) Case 2: Repeated root r. y=(C₁+C₂x)e^(rx). (7) C