Final answer:
To find a second solution of the form x(t) = tw(t) to the given differential equation when x(t) = t is known, we substitute x = tw into the equation, differentiate, and solve for w(t).
Step-by-step explanation:
To find a second linearly independent solution of the given differential equation (1 - t²)x'' - 2tx' + 2x = 0 when one solution is known, x(t) = t, we look for a solution of the form x(t) = tw(t), where w(t) is an unknown function to be determined. By substituting x = tw into the differential equation and using product rules for differentiation, we will obtain a new differential equation in terms of w and its derivatives. The goal is to simplify this new equation and find w(t) such that x(t) = tw(t) is a solution to the original equation.
After substituting into the given differential equation, we differentiate and combine like terms to arrive at an equation that only involves w(t) and its derivatives. This will likely involve simplifying terms and possibly using a reduction of order technique, which is a common method for finding a second solution to a second-order linear homogeneous differential equation when one solution is already known. Once we have found w(t), we multiply it by t to get our second linearly independent solution, thus ensuring we have a complete solution set.