I am struggling to understand what makes some differential equations well-posed Cauchy (initial value) problem rather than a Dirichlet (boundary value) problem. Consider, for example, a simple harmonic oscillator described by ODE $$\frac{d^2 x}{dt^2} + x = 0.$$ This equation has well-behaved solutions for arbitrary initial conditions $x_0 = x(t_0)$ and $v_0 = \dot{x}(t_0)$. However, if we try to impose arbitrary boundary conditions, e.g. $x_1 = x(t_1)$ and $x_2 = x(t_2)$, the solution may not be well behaved (or even exist!) if the times $t_1$ and $t_2$ differ by an integer multiple of the period $2\pi$, because $x(t + 2\pi) = x(t)$. So, what is it in the harmonic oscillator ODE that makes it a good Cauchy rather than a Dirichlet problem?

On a similar note, if we try to solve the corresponding Sturm-Liouville problem $$\frac{d^2 x}{dt^2} + \lambda x = 0,$$ for unknowns $x(t)$ and $\lambda$, then we know that this is a Dirichlet-type problem and definitely not Cauchy problem.

Is there a general theory which deals with this kind of issues?