Final answer:
To find the second solution y2(x) for the given differential equation, one must use the method of reduction of order on y1(x). This process involves assuming y2(x) is a product of y1(x) and another function v(x), and then substituting back into the differential equation to solve for v(x). However, the specific steps and computations are not provided in the question.
Step-by-step explanation:
The task is to find the second solution y2(x) for the differential equation 4x²y" - y = 0, given that one solution is y1(x) = x⁻₀.5ln(x). In problems involving second-order linear homogeneous differential equations with variable coefficients, you can often find a second linearly independent solution using the method of reduction of order. Since we are given one solution already, we can apply this method to find the complementary solution.
To apply the reduction of order, we assume a second solution of the form y2(x) = v(x)y1(x) and then substitute this into the differential equation to obtain an equation for v(x). Once we integrate this equation to get v(x), we can multiply it by the given solution y1(x) to find the second solution y2(x).
However, this specific question is part of a larger context of solving differential equations and does not provide all the details necessary to solve it completely here. To successfully find y2(x), one would need to follow the method of reduction of order step by step, which may require more detailed calculations that depend on the context of the problem.