Differential Equations
for Engineers:
the Essentials
Class 7 notes
Agenda: Class 7
Review Homework Assignment 5
Lecture: Second order linear time-invariant homogeneous ODEs:
LRC circuit
Characteristic equation
Second Order Linear Time-Invariant
Homogeneous ODEs
Example: LRC Circuit
LRC Circuit
Resistor
R ohms
Closes at t= 0
LRC Circuit (2)
Kirchhoff’s Law:
Sum of voltage drops
around a closed circuit = 0
Voltage drop over an
inductor :
Inductor
L henrys
Resistor
R ohms
Capacitor
C farads
dt
dI
LVi=
LRC Circuit (3)
Initial conditions:
ODE:
L
R
C
0)0(
)0( 0
=
=
I
VVc
0
=++
c
VRI
dI
L
LRC Circuit (4)
What is the solution to
0
1
2
2
=++ c
cc V
LCdt
dV
L
R
dt
Vd
R
rtrt
ere
dt
d
2
2
2
)(
=
Eq’n 1
LRC Circuit (5)
Substituting in Equation 1
we have
L
R
C
rt
cetV =)(
0)
1
(
2
=++
e
LC
r
L
R
rrt
LRC Circuit (6)
Solving Equation 2:
LC
r
L
R
r
0
1
2
=++
LRC Circuit: Real Roots
Case 1: Real roots
If the roots are real the solutions Equation 3 provides take the form
0
1
2
2
−
LCL
R
LRC Circuit: Real Roots (2)
To meet the initial conditions,
)(
)(
2
)(
1
+= +−−−
ecectV
tt
c
LRC Circuit: Real Roots (3)
Hence the solution to the LRC ODE in Case 1 (real roots to the
characteristic equation) is
+
+
−
−= +−−− tt
ceeVtV )()(
02
)(
2
)(
)(