Variation of parameters method
For a non-homogenous linear second-order ODE in the form:
𝑦" + 𝑝(𝑥)𝑦′ + 𝑞(𝑥)𝑦 = 𝑓(𝑥)
[NOTE]: Must be in this form, where leading term is monic. If not, rearrange.
General solution: 𝑦𝐻 = 𝐴𝑦1 (𝑥) + 𝐵𝑦2 (𝑥)
Particular solution: 𝑦𝑃 = 𝑢1 (𝑥)𝑦1 (𝑥) + 𝑢2 (𝑥)𝑦2 (𝑥)
𝑦1 𝑦2
Wronskian, 𝑊 (𝑥 ) = |𝑦 ′ 𝑦 ′| = 𝑦1 𝑦2′ − 𝑦2 𝑦1 ′
1
2
𝑢1 (𝑥 ) = − ∫
𝑢1 (𝑥)𝑓(𝑥)
𝑊(𝑥)
𝑢2 (𝑥 ) = ∫
𝑢2 (𝑥)𝑓(𝑥)
𝑊(𝑥)
Forced oscillations
𝑚𝑦̈ + 𝑐𝑦̇ + 𝑘𝑦 = 𝐹0 sin(𝜔𝑡)
Resonance: Occurs when the angular frequency of the forcing function is equal/matches to the
natural angular frequency of the SHM (homogenous system).
TOPIC #6: Matrix Theory
Systems of differential equations