LRC Circuit: Roots of
Characteristic Equation
Have finished Case 1: two real roots
Now consider Case 2: repeated real roots
LRC Circuit: Repeated Roots
Recall the solution to the LRC ODE in Case 1 (two real roots to the
characteristic equation) is
Consider this solution as approaches zero; i.e., as the two roots
approach one another.
LRC Circuit: Repeated Roots (2)
Divide Equation 7 by to obtain
t
eV
−
0
t
t
−
Equation 8 becomes
LRC Circuit: Repeated Roots (3)
Which becomes
LRC Circuit: Repeated Roots (4)
In the limit as approaches zero this becomes
To verify that Equation 9 is the solution of
LRC Circuit: Repeated Roots (5)
Note that since
and
we can write the ODE as
LRC Circuit: Repeated Roots (6)
Now
so
dt
LRC Circuit: Repeated Roots (7)
Checking initial conditions:
Hence
is a solution of the LRC ODE when
L
2
LRC Circuit: Repeated Roots (8)
The results of this example extend to any ODE of the form
The general solution is
where the coefficients are determined by initial conditions.
In-Class Example Problems
Solve:
1)0(,2)0(
065
2
2
=
=
=++
yy
y
dt
dy
dt
yd
Homework Assignment 7
Prepare for Test #1 in next class