What is the difference between the steady-state solution and the transient solution?

The steady state solution associated with the given problem. Every system, when left to itself, tends to become steady. For example it's state that becomes stable and independent of time. If we write u (×,t)- v (×)= w (×,t) then w (×,t) is called transient temperature distribution, and v (×) is called steady state temperature distribution.

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go through this link , its difference is explained with as simple example

In steady state solutions the variables of the system varies linearly (remains steady) due to the application of force with respect to time where as in transient solution with the application of force the variables of the system varies non-linearly with respect to time.

steady state is system parameter that will tend to avoid any changes in its property w.r.t time and for transient states it is vice versa

STEADY STATE is on the STATIC Condition taking mostly the effect of Pressure and Temperature, whereas in TRANSIENT STATE, on top of the PRESSURE and TEMPERATURE, the Hydrodynamics/Fluid Flow either LAMINAR or TURBULENT with respect to TIME is also considered.

If the variables of the system are not changing with time than its called steady, the solution of steady state is called steady state solution, if system variables changes with time then its transient & solution associated with it is transient solution.

In steady state the system is in equilibrium condition there will be no more phase changes, but in transient phase changes occurs until it reaches the steady state. 

As the name suggests transient response of control system means changing so, this occurs mainly after two conditions and these two conditions are written as follows-

Just after switching ‘on’ the system that means at the time of application of an input signal to the system.

Just after any abnormal conditions. Abnormal conditions may include sudden change in the load, short circuiting etc.

Steady state occurs after the system becomes settled and at the steady system starts working normally. Steady state response of control system is a function of input signal and it is also called as forced response.

Basic difference is Time.

Steady state soln is not affected with time.

and the transient soln is.

The difference between a steady state simulation and marching a transient solution to steady state is that the SS simulation ignores many of the cross terms and higher order terms dealing with time. These terms all go to zero in steady state so they don't affect the steady state result. The transient simulation includes all these terms. Usually this means the steady state model has an easier convergence as there are less terms to model and some transient non-linearities are removed, but in a few models these non-linearities help convergence (but this is infrequent).