Existence and Uniqueness theorem as it applies to finding an explicit solution If the conditions of the theorem are met for some ordinary differential equation, then we are guaranteed that a solution exists. However, I don't fully understand what it means for a solution to exist. If we can show that a solution exists, does that mean that it can be found explicitly using known methods? Or, are there some differential equations, that we know exist because of the theorem, but for which we can not find a general solution, and are thus forced to use numerical methods for an approximation?
 A: It is often the case that a differential equation (DE) can't be solved analytically, even though we can prove that a solution exists. Recall from calculus that there are even elementary functions which don't have primitives that can be written in terms of elementary functions (eg, $f(x) = e^{-x^2}$).
Having a DE that can actually be solved analytically, particularly when one is looking at partial differential equations, is actually rather special. This doesn't mean that it is worthless to know that a solution exists or to understand the qualitative properties of a solution. Such information is nice to have when investigating a DE numerically.
Edit: I'm not aware of any "commonly studied" DE that don't have solutions. However, there are certainly some famous examples of PDE with no solution. For example, an example due to Lewy says that there exists a smooth $\Bbb{C}$-valued function $F=F(z,t)$ on $\Bbb{C}\times\Bbb{R}$ such that
$$\partial_{\overline{z}}u - iz\partial_tu = F(z,t)$$
doesn't have a solution on any open set. However, examples like this are notable precisely because they don't have solutions, not because they are otherwise commonly studied.
On the other hand, there are certainly commonly studied problems that are ill-posed. Here, I'm talking about well-posedness in the sense of Hadamard (a unique solution exists and the data to solution map is continuous). A couple examples of such problems would be the backward (in time) heat equation and some inverse problems. For example, the backward heat equation is ill-posed because the solutions are extremely sensitive to the initial data (the data to solution map is not continuous).
Finally, there are many problems that are commonly studied and can't be solved analytically. In fact, I would say that this would be the case for most "commonly studied" nonlinear PDE. An example here would be the Naiver-Stokes equations. We can only solve the Navier-Stokes equations analytically in some very special cases (generally after making a number of simplifying assumptions). So, in most applications, one must solve the Navier-Stokes equations numerically.
A: Your phrasing it not vague! Indeed the Lindelöf-Picard theorem states under some conditions that a solution to the IVP problem exists and is unique.
This doesn’t mean at all that such a solution can be written using standard functions: polynomials, exponential, trigonometric... or composition of such maps.
There are cases where it can be proven that it is not possible to do so. Liouville’s theorem can be used to prove that.
