Differential Equations
for Engineers:
the Essentials
Class 7 notes
Agenda: Class 7
Review Homework Assignment 5
Lecture: Second order linear time-invariant homogeneous ODEs:
Second Order Linear Time-Invariant
Homogeneous ODEs
LRC Circuit
Resistor
R ohms
Closes at t= 0
LRC Circuit (2)
Kirchhoff’s Law:
Sum of voltage drops
Voltage drop over a resistor:
LRC Circuit (3)
Initial conditions:
ODE:
L
R
C
0)0(
)0( 0
=
=
I
VVc
0
=++
c
VRI
dt
dI
L
LRC Circuit (4)
What is the solution to
This is a 2nd order linear time-invariant
ODE. Given that the solution to a 1st order
dt
LRC Circuit (5)
Substituting in Equation 1
we have
which implies
R
rt
cetV =)(
rt
cetV =)(
R
R
r
LC
r
L
r
0
1
0
22
=
−+
+
=++
Case 1: Real roots
Hence the general solution to the LRC ODE for Case 1 is
0
1
2
2
−
LCL
R
tt
cecectV )(
2
)(
1
)(
+−−− +=
Equation 4
LRC Circuit: Real Roots (2)
dt
Solving Equations 5 and 6 we find that
01
2
)(
Vc
−
−=
2
L
R
2
=
R1
2
−
=
where