Differential Equations
for Engineers:
the Essentials
Supplement to Class 4 Notes
Contents of Supplement
Differences Between Linear and Nonlinear ODEs in Their Input / Output
Response
Differences Between Linear and Nonlinear
ODEs in Their Input / Output Response
Input / Output Characteristics
for a Linear System
In a linear system of any order (time-varying or time-invariant):
If the output is when the input is and the output is
)(
1ty
)(
)(
In a linear time-invariant system of any order, after transients have
died away, when the input is a sine-wave of a given frequency
Existence and Uniqueness of
Solutions to Linear ODEs
Theorem: Given a linear nth order ODE:
with initial conditions
(Stated without proof.)
Key Points from the
Following Nonlinear Example
The method of successive approximations (solving a sequence of
linear equations to approximate the solution of a nonlinear
Nonlinear Example: LR Circuit
LR
Kirchhoff’s Law:
Sum of voltage drops around a
closed circuit = 0
Voltage drop over an inductor:
Voltage drop over a resistor:
Resulting equation :
Nonlinear Example: LR Circuit (2)
The solution to
)/arctan(
1
=
LR/=
is
where
Nonlinear Example: Recalling the LR Circuit (3)
Input / output response:
The “input” to the “system” is the voltage source. The “output” is
the current (or voltage) over the resistor.
What if the Resistor is Slightly Nonlinear?
Instead of
suppose we have
What if the Resistor is Slightly Nonlinear? (2)
Approach to Solution
of Slightly Nonlinear ODE
Since is small, consider the successive approximations
0)0(
)sin(
1
01
1
=
=+
I
tII
dt
dI
Equation 2
Approach to Solution
of Slightly Nonlinear ODE (2)
We are solving the nonlinear Equation 1 approximately by the sequential
solution of a series of linear Equations 2, 3 and 4.
The general solution of
Using Equation 5, we have already found that the solution to Equation (2) is
Approach to Solution
of Slightly Nonlinear ODE (3)
0
Repeating Equation 3:
This has solution
Now
Approach to Solution
of Slightly Nonlinear ODE (4)
Continuing in this way, we find
which one can show can be written as
where
Note the higher frequencies –the harmonics
Approach to Solution
of Slightly Nonlinear ODE (4)
Challenge problem:
Given the steady state solution