Chapter 4.1, Problem 33E

(a)

To determine

A second linearly independent solution of the differential equation $t^2{y}^{″}+t{y}^{\prime }+\left(t^2-\frac{1}{4}\right)y=0$ by the use of reduction of order with the solution $y_1\left(t\right)=\frac{\cos t}{\sqrt{t}}$.

(b)

To determine

The general solution of the differential equation $t^2{y}^{″}+t{y}^{\prime }+\left(t^2-\frac{1}{4}\right)y=0$.

(c)

To determine

The fundamental set of solutions to the differential equation $t^2{y}^{″}+t{y}^{\prime }+\left(t^2-\frac{1}{4}\right)y=0$.

(d)

To determine

The solution to the initial value problem $t^2{y}^{″}+t{y}^{\prime }+\left(t^2-\frac{1}{4}\right)y=0$ for the given initial conditions.