About PDE's: always write the initial part as a Fourier series sum_{n} b_{n} sin(n x):
Sum b_{n} exp(-n^{2} t) sin(n x) solves heat u_{t} = A u = u_{xx} for position
Sum b_{n} cos(n t) sin(n x) solves wave u_{tt} = A u = u_{xx} for position
Sum b_{n} sin(n t)/n sin(n x) solves wave u_{tt} = A u = u_{xx} for velocity
Proof: sin(n x) is the eigenfunction of the eigenvalue L = -n^2 of A=D^2.
With sin(n x) as initial heat we get u_t = L u solved by e^(-n^2 t).
With sin(n x) as initial position, get u_tt +n^2 u = 0 solved by cos(nt).
with sin(n x) as initial velocity, get u_tt +n^2 u = 0 solved by sin(nt)/n
This can even fit into a song.
(More generally, just replace -n^2 t with L t and n t with sqrt(-L) t,
where L is the eigenvalue of A to the eigenfunction sin(nx).
Some slides from the review with comments.