An important class of continuous time LTI systems are those modeled by ordinary linear differential equations with constant coefficients.

Given an input *x*(*t*) and an
output *y*(*t*),

is a linear first-order (the
order of the derivatives is one)
differential equation with constant coefficients (as long as *a* and *b* are constants).

A general *n*th order linear differential equation with constant
coefficients is:

which we can write as:

A classical method for the solution of our differential equation is
called the *method of undetermined coefficients*. We express the output *y*(*t*) as the
sum of *complementary* or *natural* (*y*_{c}(*t*)) and
*particular* or *forced* (*y*_{p}(*t*)) solutions:

*y*(*t*) = *y*_{c}(*t*) +
*y*_{p}(*t*)

**Natural response** The natural response
*y*_{c}(*t*) is the solution to the homogeneous equation:

Assume that the solution of the homogeneous equation is:

*y*_{c}(*t*)= *Ce*^{st}

Substituting in the homogeneous equation yields:

and we get:

This is the *characteristic equation* and it may be factored as:

The solution is of the form:

assuming there are no repeated roots (which is all we will cover here).

__Ex.__ Given a first-order differential equation

find its homogeneous solution. Your answer should be in terms of a constant
*C*.

**Forced response** The forced response *y*_{p}(*t*)
solves the equation

The form of the solution is determined by the input *x*(*t*).
For an exponential input
*x*(*t*)= *A e*^{at}, the solution would be
*y*_{p}(*t*) = *P e*^{at} where *A*, *a*, and
*P* are constants.

__Ex.__ For the previous example, assume an input
*x*(*t*) = 6 *e*^{3 t}. Find the particular solution
*y*_{p}(*t*).

__Ex.__ Now, assuming that the system is initially at rest (the initial conditions are 0),
i.e., *y*(0) = 0 ,
solve for the constant *C* in your
overall solution
*y*(*t*) = *y*_{c}(*t*) + *y*_{p}(*t*):

__Ex.__ Given

*y* '(*t*) + 2*y*(*t*) = *x*(*t*)

where
*x*(*t*) = 4 *e*^{2t}, find *y*(*t*). Assume *y*(0) = 0.