Final answer:
To solve the differential equation y'' + x²y = 0 using power series methods at the point x₀ = 0, assume a power series solution of the form y(x) = ∑[n=0]∞ aₙxⁿ. Plug this into the differential equation, differentiate twice, and substitute the power series into the equation. Equate the coefficients of like powers of x to zero and solve the recurrence relation for the coefficients aₙ.
Step-by-step explanation:
To solve the differential equation y'' + x²y = 0 using power series methods at the point x₀ = 0, we can start by assuming a power series solution of the form y(x) = ∑[n=0]∞ aₙxⁿ. Plugging this into the differential equation, we can differentiate twice and substitute the power series into the equation. By equating the coefficients of like powers of x to zero, we can determine the recurrence relation for the coefficients aₙ. Solving this recurrence relation will give us the power series solution for y(x).