Final answer:
The process to find a second linearly independent solution to the provided differential equation involves assuming a form for the second solution based on a known first solution and substituting it into the original equation. However, without more information or the first solution, the specific process cannot be completed.
Step-by-step explanation:
Finding a Linearly Independent Solution
To find a second linearly independent solution to the differential equation (x²−1)y″−2xy′+2y=0, we typically use a method such as reduction of order if one solution is known, or explore special functions that are solutions to this type of equation, such as Legendre polynomials for equations with the form (1-x²)y″−2xy′+n(n+1)y=0, where n is a nonnegative integer. However, based on the information provided, it seems we lack enough data to directly identify a specific solution or method. What we can do is describe the general approach to finding such a solution.
Given one solution, let's denote it as y₁(x), we can look for a second solution, say y₂(x), which can typically be found by assuming a form y₂(x) = v(x)y₁(x) and then substituting this assumed solution into the original equation. We then solve for v(x), usually resulting in a simpler differential equation. By obtaining v(x), we can then find y₂(x). It's crucial to ensure that the second solution is not just a scalar multiple of the first, hence the term independent solution.
Without the specific first solution or additional information, the process cannot be completed here. However, the general method described would serve as a guideline for the student to follow in attempting to find the independent solution.