initial conditions, pde Together with a differential equation and initial conditions arise the notion of a well-posed problem. My question is, how does one choose the type of initial conditions to be implemented in the problem or is it due to the nature of the differential equation? Any reference to this answer will be welcomed.

To be specific, let us consider the differential equation $x''(t)+x(t)=0$ for the SHM; for which in order to start oscillations you need to release the mass bob (tied to a thin thread with other end clamped) from extreme left i.e. $x(0)=-a$. Moreover, assuming that no external force is provided at the start i.e. $x'(0)=0$.
Edit: In case of PDE, considering the well known example of wave-equation is useful.
$$\frac{1}{c^2}\frac{\delta^2y}{\delta t^2}=\frac{\delta^2y}{\delta x^2}$$
where $c$ is a constant, $y(x,t)$ is the displacement along $Y-$ axis at position $x$  on $X-$ axis and time $t$. The initial conditions are as follows:
$1$. No displacement at the ends of string (of length l) at any time $t$ i.e. $y(0,t)=y(l,t)=0$.
$2$. No initial velocity at any point $x$ in the beginning $t=0$ i.e. $\frac{\delta y}{\delta t}(x,0)=0$
$3$. To initiate the vibrations in the string, suppose the string is stretched in the form of curve $y=f(x)$ i.e. $y(x,0)=f(x)$.
