Define power series; solve y'+2xy=0

Power series

An infinite series of the form

Σ aₙ(x − x₀)ⁿ = a₀ + a₁(x − x₀) + a₂(x − x₀)² + …

(with constant coefficients aₙ) is called a power series about the point x₀. About x₀ = 0 it is Σ aₙxⁿ. It converges for |x − x₀| < R, where R is the radius of convergence.

Solve y′ + 2xy = 0

x = 0 is an ordinary point. Let y = Σ aₙxⁿ, so y′ = Σ (n+1)aₙ₊₁ xⁿ and 2xy = Σ 2aₙ₋₁ xⁿ (n ≥ 1).

Coefficient of x⁰: a₁ = 0. Coefficient of xⁿ (n ≥ 1): (n+1)aₙ₊₁ + 2aₙ₋₁ = 0, so

aₙ₊₁ = − 2aₙ₋₁ / (n + 1)

a₁ = 0  →  all odd coefficients a₃ = a₅ = … = 0
a₂ = −2a₀/2 = −a₀
a₄ = −2a₂/4 = a₀/2
a₆ = −2a₄/6 = −a₀/6

Pattern: a₂ₘ = (−1)^m a₀ / m!.

y = a₀(1 − x² + x⁴/2! − x⁶/3! + …) = a₀ e^(−x²)

Answer: y = a₀e^(−x²). (Check by separation of variables: dy/y = −2x dx gives y = Ce^(−x²).)