Chapter 13.2, Problem 5P

Show that the solutions of (2.5) can also be written as

Also show that these solutions are equivalent to (2.7) if $k$ is real and equivalent to (2.18) if k is pure imaginary. (See Chapter 2, Section 12.) Also show that $X=\sin k(x-a),Y=\sinh k(y-b)$ are solutions of (2.5)..