There are only few forms of such equations that can be explicitly solved (for example Euler's equation)
Other equations can not be solved directly, but they are important enough to get their own names and to be studied with different tools to give some characteristics of them (asymptotic behaviour, integral and differential identities, infinite sum expansion) but cannot be explicitly written (for example Bessel's Equation and its solutions the Bessel Functions)
You can always know the second solution of an ODE by just knowing the first solution, so from times to times guessing can come in handy.
In general I believe there are many problems to study that arise from second order linear ODE, and you can acquire many tools to do so along the way. But the bottom line is that there is no general method of solving them.