What do we actually mean when we say that a certain problem has or does not have an analytical solution? I ask this because some systems that are said to have an analytical solution actually are no better than some systems that are said not to have it. I will give two examples. The standard harmonic oscillator, whose equation of motion is $$m\ddot{x}=-kx$$, has the general solution $$x(t)=A\cos(\omega t+\phi)=A\cos\delta(t),$$ where $$\omega=\sqrt{\frac{k}{m}}$$ and $\delta(t)=\omega t + \phi$. However, that cosine function is defined as $$\cos\delta=\sum_{n=0}^\infty \; (-1)^n \; \frac{\delta^{2n}}{(2n)!}.$$ Which can only be calculated approximately. How is this different with the simple pendulum, where the solution is defined in terms of Jacobi elliptic functions, and can also be calculated only approximately?
Another example, the free particle with linear drag. The equation of motion is $m\dot{v}=-bv$. The solution with initial velocity $u$ is $v(t)=ue^{-bt/m}$. Here, $e$ is an irrational number and thus the velocity cannot be calculated exactly.
Moreover, for any complicated problem, we may define a function to be the solution of the problem, and may be tabulated.
So, why do we say the harmonic oscillator has an analytic solution but the pendulum not? What makes a solution analytical? Also, what means physically that a system has no analytical solution?