Final answer:
To solve the DE y" - xy - y = 0 using power series, one assumes a power series solution, derivatives term by term, and then finds a recurrence relation for the coefficients by equating like powers of x. The solution will typically involve Airy functions.
Step-by-step explanation:
To solve the differential equation y" - xy - y = 0 using the method of power series, we first assume a power series solution of the form:
y = ∑ a_n x^n, where n starts from 0 going to infinity.
Consequently, we differentiate term by term to find y" and y'. By plugging y, y', and y" into the original differential equation, we can obtain a recurrence relation for the coefficients a_n. Using the standard comparison of coefficients for power series, we set the coefficients of like powers of x to be equal on both sides of the equation, which enables us to find a pattern or formula for the a_n's. For this particular differential equation, special functions known as Airy functions typically arise, and the general solution will be a combination of these functions.
However, since the question does not provide specific initial conditions or request to find specific coefficients, we won't be able to provide a more detailed solution at this moment. To fully solve the problem, one would follow the steps outlined above to arrive at the general solution.