Final answer:
To employ the reduction of order method for the given second-order linear homogeneous differential equation with a known solution, a second solution is constructed by multiplying the first solution by an unknown function and substituted back into the original equation to find a simpler equation for the unknown function, which is then solved.
Step-by-step explanation:
To find a second solution to the differential equation t² y′′ + ty′ + (t² - ⅔)y = 0 using the method of reduction of order, we need to leverage the given first solution y1(t) = t^{-1/2} sin(t).
We look for a second solution of the form y2(t) = v(t)y1(t). We then substitute y2(t) into the original differential equation and use the fact that y1(t) satisfies the equation to simplify and find an equation for v(t).
Through this process, we typically derive a simpler differential equation for v(t) which can often be solved through integration. Once we find v(t), we obtain y2(t) by multiplying v(t) by the known solution y1(t). The particular steps and integrations involved would depend on the specifics of the differential equation and the function v(t) that emerges.