Unfortunately, Theorem 1.23 does not show us how to find two independent solutions. Howeve...

Unfortunately, Theorem 1.23 does not show us how to find two independent solutions. However, there is a technique that can be used to find a second solution when one solution is known.

(a) Show that y1 (t) = t2 is a solution of

(b) Let y2(t) = υy1(t) = υt2, where υ is a yet to be determined function of t. Note that if y2/y1 = υ and υ is nonconstant, then y1 and y2 are independent. Show that the substitution y2 = υt2 reduces equation (1.32) to the separable equation

Solve equation (1.33) for v, form the solution y2 = υt2, and then state the general solution of equation (1.32).