Unit Step Response of a System

The unit step response of a system is the output of the system when the input is a unit step function. In this case, the system is described by the differential equation:

dy(t)/dt = 3y(t) + x(t)

Where dy(t)/dt is the derivative of y(t) with respect to time t, y(t) is the output of the system, and x(t) is the input to the system.

To find the unit step response, we need to consider the input as a unit step function, which can be defined as:

x(t) = 1 for t >= 0
x(t) = 0 for t < />

The differential equation can now be rewritten as:

dy(t)/dt = 3y(t) + 1

Solving the Differential Equation

To solve the above differential equation, we can use the method of integrating factors. The integrating factor is given by:

IF = e^(∫3dt) = e^(3t)

Multiplying both sides of the differential equation by the integrating factor, we get:

e^(3t) * dy(t)/dt = 3e^(3t) * y(t) + e^(3t)

Integrating both sides with respect to t, we get:

∫e^(3t) * dy(t)/dt dt = ∫(3e^(3t) * y(t) + e^(3t)) dt

Using the property of integration, we can simplify the equation as:

e^(3t) * y(t) = ∫3e^(3t) * y(t) dt + ∫e^(3t) dt

Solving the integrals, we get:

e^(3t) * y(t) = ∫3e^(3t) * y(t) dt + (1/3)e^(3t) + C

Where C is the constant of integration.

Applying the Initial Condition

To find the unit step response, we need to apply the initial condition. Since the system is at rest initially, the output y(0) is zero.

Now, substituting y(0) = 0 into the equation, we get:

e^(3t) * 0 = ∫3e^(3t) * 0 dt + (1/3)e^(3t) + C

Simplifying the equation, we get:

0 = 0 + (1/3)e^(3t) + C

Therefore, C = - (1/3)e^(3t)

Final Expression for the Unit Step Response

Substituting the value of C into the equation, we get:

e^(3t) * y(t) = ∫3e^(3t) * y(t) dt + (1/3)e^(3t) - (1/3)e^(3t)

Simplifying the equation, we get:

e^(3t) * y(t) = ∫3e^(3t) * y(t) dt

Dividing both sides by e^(3t), we get:

y(t) = ∫3y(t) dt

Now, taking the Laplace transform of the equation, we get:

Y(s) = 3Y(s)/s

Solving the equation for Y(s), we get:

Y(s) = 3Y(s)/