A linear system adheres to the principles of superposition and homogeneity, represented by linear equations. Time-variant systems have outputs that depend on time, while time-invariant systems do not. The transfer function for linear time-invariant systems expresses the output-to-input ratio via Laplace transforms when initial conditions are zero, serving to analyze system responses.

1. Linear system
• A system satisfies the principal of supper
position and homogeneity is called linaer
system.
• The system can be represented by set of linear
equation.
• Y(t)=mx(t)

2. Time variant system
• Time variant system: The system’s output
directly depends on time.
• Y(t)=tx(t)
• Time invariant system: The system’s output
does not directly depend on time.
• Y(t)=ax(t)

3. Transfer function
For linear time invariant system the transfer
function is defined as the ratio of Laplace
transfer of output to Laplace transform of
input under the assumption that all initial
condition are zero.

4. Characteristic of transfer function
• It relates output variable to input variable.
• It is the property of system itself.
• It does not provide the physical structure of the
system.
• It is used to analysis and determine the response
of the system under different input.

5. Steps to find transfer function
• Find the mathematical equation depending
upon the physical law.
• Take Laplace transform transformation of the
equations considering the initial condition
zero.
• Find the output to input ratio.

6. Laplace transform of the equation

7. Laplace transform of the equation