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As nouns the difference between vagrant and transient is that vagrant is a person without a home or job while transient is something which is transient. As adjectives the difference between vagrant and transient is that vagrant is moving without certain direction; wandering; erratic; unsettled while transient is passing or disappearing with time; transitory.
The contrast of transient is steady state. Steady state refers to a situation whereby which the system has settled such that its parameters are seemingly constant. Transient means conditions have most suddenly activated and all portions of the system are responding to the initial conditions. The general solution of a differential equation is an equation with several arbitrary parameters, each of which will become known once initial conditions and boundary conditions are known.