Line of Solution: If p̅ and h̅ are vectors in vector space V, with h̅ ( 0̅, then the line through p̅ in the direction h is defined to be the set (x̅ ( V | x̅ = p̅ + th̅ t ( R}.
(a) Find the line in R2 through p̅ = (0, 1) in the direction h̅ = (2, 3).
(b) Find the line in R3 through p̅ = (2, 12) in the direction h̅ = (2, -3. 0).
(c) Show that solutions of y' + y = 0 are a subspace of V = C1 (-(, (), and that every vector in this sub-space is a multiple of e-t.
(d) Show that the solutions of y' + y = t form a line in V through t - 1 in the direction e-t.
(e) Relate parts (c) and (d) to what you learned about solutions to homogeneous and non-homogeneous differential equations in Sec. 2.1.