(a) Use implicit differentiation to show that $t^{2}+y^{2}=C^{2}$ implicitly defines solutions of the differential equation $t+y y^{\prime}=0$.
(b) Solve $t^{2}+y^{2}=C^{2}$ for $y$ in terms of $t$ to provide explicit solutions. Show that these functions are also solutions of $t+y y^{\prime}=0$.
(c) Discuss the interval of existence for each of the solutions in part (b).
(d) Sketch the solutions in part (b) for $C=1,2,3,4$.