Differential equations, zero state response

[serp777]

Homework Statement

Find the zero input and zero state response for the following system

y''(t) + 3y'(t) + 2y(t) = 2 x'(t) - x(t-1)

where x(t) = (2e^-t)*u(t)

U(t) is the step function

Homework Equations

Y = Yh + Yp

Y = Yzsr + Yzir

The Attempt at a Solution

I can't find any similar examples online and I am partially thrown off by the u(t) step function, and it's derivative the ζ(t) function.

I have no issues finding the homogenous equation, but the particular part is confusing, specifically finding coefficients. There are no table forms I can find that I can plug back into the differential equation to solve for.

To start finding the particular form, I used the product rule with the step function

right hand side of equation = (-4*e^-t)*u(t) + (4*e^-t)*ζ(t) - 2*e^-(t-1)*u(t)

But I have no idea how to solve for coefficients of this system, basically stuck and I am not able to find the zero state response without finding the particular form.

 

[vanhees71]

It's a bit a strange language. The "zero-input response" seems to be the general solution of the homogeneous equation,
$$y''+3y'+2y=0,$$
and the "zero-state response" the particular solution of the inhomogeneous equation with the homogeneous initial conditions $y(0)=y'(0)=0$. I think, this makes the problem much clearer.

The homogeneous problem is solved by the standard ansatz
$$y(t)=C \exp(\lambda t)$$
by figuring out the two possible values for $\lambda$ and write down the most general solution in terms of the general superposition of the corresponding two solutions.

The inhomogeneous equation can be solved with help of the Green's function, which seems to be easier to find than to directly solve the problem with the given inhomogeneity.

Finally, note that
$$u'(t)=\delta(t),$$
where $\delta$ denotes the Dirac-$\delta$ distribution.