For the circuit shown,

show that if R1=R2=C=1 that

The circuit shown (with R1=R2=C=1) is characterized by the differential equation shown

Assume e_{in}(t)=0 for t≤0. If e_{out}(0-)=0 and e_{in}(t) =
6·t^{2} + 2 (for t≥0):

- Find the form of the homogeneous response.
- Find the form of the particular response.
- Find the unknown coefficients in the particular response.
- Find e
_{out}(t).

a)

b) Particular solution is same form as input and its derivatives

c)

d)

The circuit shown (with R1=R2=C=1) is characterized by the differential equation shown

Assume e_{in}(t)=0 for t≤0. If e_{out}(0-)=0 and e_{in}(t) = 8·e^{2t} (for
t≥0):

- Find the form of the homogeneous response.
- Find the form of the particular response.
- Find the unknown coefficients in the particular response.
- Find e
_{out}(t).

a)

b) Particular solution is same form as input and its derivatives

c)

d)

The circuit shown (with R1=R2=C=1) is characterized by the differential equation shown

Assume e_{in}(t)=0 for t≤0. If e_{out}(0-)=0 and e_{in}(t) =
8·e^{-2t} (for
t≥0):

- Find the form of the homogeneous response.
- Find the form of the particular response.
- Find the unknown coefficients in the particular response.
- Find e
_{out}(t).

a)

b) Particular solution is same form as input and its derivatives, but in this case the input is the same form as the homogeneous response, so we must multiply it by t.

c)

d)

Consider the differential equation

a) The particular solution is the same form as the input and represents
the solution as t→∞. In this case we have a constant
input, so y_{p}(t)=K. What is the value of K?

b) For first order constant coefficient differential
equations we assume a homogeneous solution (RHS of
equation = 0) of y_{h}(t)=Ae^{st}. What value of
's' satisfies the differential equation?

c) The complete solution is given by y(t)=y_{p}(t)+y_{h}(t).
If we have y(0^{+})=1, determine the value of the constant A
in y_{h}(t).

Note: y(0^{+}) is the value of y(t) at t=0^{+},
i.e., an infinitesimally small time after t=0.

d) What is y(t)?

a) Put y_{p}(t)=K into differential equation and use the fact that
it's derivative is zero.

b) Put y_{h}(t)=Ae^{st} into differential equation, set
the RHS to zero, and solve for s

c)

d)

Consider the differential equation

a) The particular solution is the same form as the input and represents
the solution as t→∞. In this case we have a constant
input, so y_{p}(t)=K. What is the value of K?

b) For constant coefficient differential equations we
generally assume a homogeneous solution (RHS of
equation = 0) of y_{h}(t)=Ae^{st}. What value(s) of s
satisfy the homogeneous differential equation? What is the form of the
homogeneous response?

c) The complete solution is given by y(t)=y_{p}(t)+y_{h}(t).
Determine the value of the constant(s) A in y_{h}(t) if

d) What is y(t)?

a) Put y_{p}(t)=K into differential equation and use the fact that
it's derivative is zero.

b) Put y_{h}(t)=Ae^{st} into differential equation, set
the RHS to zero, and solve for s

c)

d)

Consider the differential equation

a) The particular solution is the same form as the input and represents
the solution as t→∞. In this case we have a constant
input, so y_{p}(t)=K. What is the value of K?

b) If we
assume a homogeneous solution (RHS of
equation = 0) of y_{h}(t)=Ae^{st}. What value(s) of s
satisfy the homogeneous differential equation? What is the form of the
homogeneous response?

c) The complete solution is given by y(t)=y_{p}(t)+y_{h}(t).
Determine the value of the constant(s) A in y_{h}(t) if

d) What is y(t)?

a) Put y_{p}(t)=K into differential equation and use the fact that
it's derivative is zero.

b) Put y_{h}(t)=Ae^{st} into differential equation, set
the RHS to zero, and solve for s. There is only one value of "s" so
system is critically damped. Use appropriate form of homogenous
equation (i.e., time multiplied exponential).

c)

d)

When solving a differential equation such as :

where y(t) is the unknown function and f(t) is a known input:

- What does the particular response represent?
- Given an input function, how is the form of the particular response chosen?
- What does the homogeneous response represent?
- How is the homogeneous response chosen?
- For this third order example, what are possible forms of the
homogeneous response?

For example, a second order system has three possible forms:- If there are two distinct, real roots (s
_{1}, s_{2}) the solution is of the form:

- If there is a single real root, repeated once
(s
_{1}) the solution is of the form:

- If there is a pair of complex conjugate roots
(s
_{1,2}=-σ±jω) the solution is of the form:

- If there are two distinct, real roots (s

- The particular response is the response due to the input as t→∞ (i.e., after all the transients have died out).
- The form of the particular response is based on the form of the input function, f(t). It is essentially a guess (the "uniqueness theorem" states that only a single solution to such a problem exists, so if we guess one that works, it must be correct).
- The homogeneous response is the characteristic response of the system when there are no inputs. It has no real physical meaning, but is mathematically useful.
- The homogeneous response is also a guess (that happens to work).
We assume yh(t)=Ae
^{st}, and we set f(t)=0. This yields a third order polynomial in s. - There are several possible combinations: