Question

Use the reduction of order procedure to determine a second linearly independent solution. Given that y₁(t) = t⁻¹ is a solution to t²y'' + 3ty' + y = 0, t > 0.

Answer 1

To find a second linearly independent solution, we assume y₂(t) = v(t)y₁(t) and substitute it into the given differential equation t²y'' + 3ty' + y = 0. Solving for v(t), we find v(t) = -1/t.

Step-by-step explanation:

To find a second linearly independent solution using the reduction of order procedure, we start with the given solution y₁(t) = t⁻¹. We assume a second solution of the form y₂(t) = v(t)y₁(t), where v(t) is an unknown function. We substitute this into the differential equation and solve for v(t). In this case, we substitute y₁(t) = t⁻¹ into the equation t²y'' + 3ty' + y = 0 to get t²v'' + 3tv' + v = 0. By simplifying and solving, we find that v(t) = -1/t.