A sixth order backward differentiation formula (BDF) need six (five plus initial value) previous solutions to get started. How I can get these previous solutions? I need a method accurate to sixth order capable of handling stiff problems.
Just to add a bit to @Jesse Chan's answer: you can preserve 6th-order convergence if you use a 5th-order starting method; in general, the starting method can be one order lower than the multistep method without decreasing the global convergence rate.
The most common starting methods are, of course, Runge-Kutta methods.