The solution of a linear first-order ODE, $y'+P(x)y+Q(x)=0$, is expressible by integrals involving elementary functions, $P(x)$ and $Q(x)$. This can be proved e.g. by the applying the method of integrating factor to the equation.

This makes me wonder - are solutions to the general *second*-order ODE expreesible using integrals? We can concentrate only on homogenous equations - the inhomogenous case is accounted for by the method of variation of parameters. I could not find a similar formula in this case, so I suspect that none exists. However, even equations such as Bessel's equation admit solutions expressible by integrals, e.g. $J_n(x)=\frac 1 \pi \int_0^\pi \cos(nt-x\sin t) \,\text{d}t$. Is such a solution possible for every linear second-order ODE?