Question

Use the power series method to solve the given initial-value problem. (Format your final answer as an elementary function.) (x−1)y′′ −xy ′ +y=0,y(0)=−4,y ′ (0)=2 y=

Answer 1

To solve the given initial value problem using the power series method, assume that the solution has a power series representation y(x) = Σ(a_n * x^n). Differentiate y twice, substitute it into the differential equation, equate the coefficients, solve the recurrence relation, and use the initial conditions to determine the values of the coefficients.

Step-by-step explanation:

To solve the given initial value problem using the power series method, we assume that the solution has a power series representation y(x) = Σ(a_n * x^n), where a_n is the n-th coefficient of the power series. Differentiating y twice and substituting it into the differential equation, we can equate the coefficients of like powers of x. This results in a recurrence relation for the coefficients. Solving the recurrence relation and substituting the initial conditions, we can determine the values of the coefficients, which can be used to find the power series representation of y(x).