Use the method of reduction of order to find a second solution y2 of the given differential equation such that {y1, y2} is a fundamental set of solutions on the given interval.

t ² y′′ +2ty′ −2y=0, t > 0, y1(t)=t

(a) Verify that the two solutions that you have obtained are linearly independent.

(b) Let y(1) = y0, y′(1) = v0. Solve the initial value problem. What is the longest interval on which the initial value problem is certain to have a unique twice differentiable solution?

(c) Now suppose that y(0) = y0, y′(0) = v0. Is there a solution in this case (how does it depend on y0 and v0? If so, how many? Explain.