Analysis of BJT transistor - is a linear analysis possible?
Transistors are fundamentally nonlinear devices, so no, a strictly linear analysis is not possible.

This is why circuit simulators like SPICE were developed. They take two approaches to make the problem tractable:


For small-signal analysis, you linearize the nonlinear equations around the operating point. This works as long as the deviations of the signal from the operating point cause negligible errors relative to the nonlinear equations.
For large-signal transient analysis, you linearize the equations around the current state, and pick a time step that is small enough that the deviations from the nonlinear equations are negligible. If you find that currents or voltages are changing "too fast" (a settable parameter), you reduce the size of the time step.