Question

Consider the third order LTI system: Assume, y(0)−5y(t)+6y(t)−2y(t)=3f(t)−2f(t),t>0 Assume, y(0⁻ )=1, y˙(0⁻ )=0,y(0⁻ )=0 and f(t)=e−ᵗ uᵗ. Solve for h(t), yₛₜₑₚ (t), yₛₛ (t) and yₙ (t). Employ the taplace transform method.

Answer 1

The question involves solving a third-order LTI system using the Laplace transform to find the impulse, step, steady-state, and natural responses for an input exponential decay function.

Step-by-step explanation:

The question relates to solving a third-order linear time-invariant (LTI) system using the Laplace transform method. The equations and initial conditions specified aim at finding the system’s impulse response h(t), the step response y_step(t), the steady-state response y_ss(t), and the natural response y_n(t) when the input f(t) is an exponentially decaying function multiplied by the unit step function, e-t u(t). To solve for these, one would need to apply the Laplace transform to both sides of the given differential equation, use algebraic manipulation to solve for the Laplace-transformed output Y(s), and then apply initial conditions to find the complete response. Afterward, the inverse Laplace transform is used to obtain the time domain functions for h(t), y_step(t), y_ss(t), and y_n(t).

It is important to note the differential equation may contain a typo as it does not make sense in its current form. The typical form of a linear differential equation would include derivatives of the function y(t). The relevant parts of the question also suggest an understanding of superposition and wave equations from the field of physics.