Final answer:
To solve the initial-value problem using a power series, we assume y as a sum of aₙxⁿ, differentiate to get y' and y'', substitute into the original equation, and equate coefficients to find a recursive relation for aₙ.
Step-by-step explanation:
The task involves solving the initial-value problem where the second derivative of y minus three times the product of the variable x and the first derivative of y minus three times y itself equals zero. Given initial conditions are y(0) = 1 and y'(0) = 0. The solution is assumed to be represented by a power series in the form y=∑ₙ₌₀[infinity] aₙxⁿ. To find coefficients aₙ, one can differentiate the power series term by term to obtain expressions for y' and y'', substitute them into the differential equation, and equate coefficients of like powers of x to find a recursive relation for aₙ.
We start with y=∑ₙ₌₀[infinity] aₙxⁿ. The first derivative, y', will be ∑ₙ₌₁[infinity] naₙxⁿ⁻¹ and the second derivative, y'', will be ∑ₙ₌₂[infinity] n(n-1)aₙxⁿ⁻². Substituting these into the differential equation and comparing the coefficients for each power of x will yield a set of equations that can be used to recursively solve for the coefficients aₙ with respect to the given initial conditions.
The method to solve for these coefficients involves setting up a system of equations based on the initial conditions y(0) = 1 and y'(0) = 0, and the recursion that arises when the power series and its derivatives are plugged into the original differential equation. This is a standard method in differential equations known as the power series method.
In summary, the power series method is commonly used to approach such problems when the function is believed to be representable by a power series expansion and particularly in cases where other methods are not suitable or more complicated to implement.