Final answer:
To solve the initial value problem, a recurrence relation for the coefficients an is derived from the differential equation and the initial conditions, allowing the calculation of terms in the series up to the fourth degree.
Step-by-step explanation:
The question pertains to finding the coefficients in the series expansion of the solution to a second-order linear differential equation. Given the initial value problem y'' + 6xy' + 10y = 0; y(0) = 1; y'(0) = 0, we aim to derive a recurrence relation for the coefficients an and calculate the terms up to a4. To develop the series expansion, we assume a solution of the form y = ∑ (anxn), where n goes from 0 to infinity. By differentiating this assumed solution, substituting into the given differential equation, and equating coefficients of like powers of x, we find the recurrence relation.
Using the initial conditions y(0) = 1 and y'(0) = 0 helps us determine the first few coefficients. The calculation involves systematic derivation of each term based on the previous ones, according to the derived recurrence relation, resulting in finding the values for a0 through a4.