notesonly.in

Q: Solve y'-2y = 0 by power series about 0. y = Σaₙxⁿ. y' = Σnaₙxⁿ⁻¹ = Σ(n+1)aₙ₊₁xⁿ. Σ[(n+1)aₙ₊₁-2aₙ]xⁿ = 0: (n+1)aₙ₊₁ = 2aₙ. aₙ = 2ⁿa₀/n!: y = a₀Σ(2x)ⁿ/n! = a₀e^(2x)! (Power series rediscovers exponential!)