Suppose that p is odd and q is even (i.e., p(−x) = −p(x), q(−x) = q(x) for all x).

(a). If y is a solution to (HE) on [−a, a], show that ye(x) = y(−x) is also a solution.

(b). Deduce that the solution to (HE) with y(0) = 1, y'(0) = 0 is even.
Hint: Show that 1/2 (y(x) + y(x)) is also a solution obeying the same conditions at x = 0 as y, and use the result on uniqueness of solution.

(c). What can be said about the solution to (HE) with y(0) = 0, y'(0) = 1?

Question

Suppose that p is odd and q is even (i.e., p(−x) = −p(x), q(−x) = q(x) for all x).

(a). If y is a solution to (HE) on [−a, a], show that ye(x) = y(−x) is also a solution.

(b). Deduce that the solution to (HE) with y(0) = 1, y'(0) = 0 is even.
Hint: Show that 1/2 (y(x) + y(x)) is also a solution obeying the same conditions at x = 0 as y, and use the result on uniqueness of solution.

(c). What can be said about the solution to (HE) with y(0) = 0, y'(0) = 1?