Chapter 50
Second order differential
equations of the form
a
d 2 y
dx 2 + b
dy
dx + cy = 0
50.1 Introduction
An equation of the form a
d 2 y
dx 2 + b
d y
dx
+ cy = 0, where
a, b and c are constants, is called a linear second
order differential equation with constant coefficients. When the right-hand side of the differential
equation is zero, it is referred to as a homogeneous
differential equation. When the right-hand side is not
equal to zero (as in Chapter 51) it is referred to as a
non-homogeneous differential equation.
There are numerous engineering examples of second
order differential equations. Three examples are:
(i) L
d 2 q
dt 2 + R
dq
dt
+
1
C
q = 0, representing an equation for charge q in an electrical circuit containing
resistance R, inductance L and capacitance C in
series.
(ii) m
d 2 s
dt 2 + a
ds
dt
+ ks = 0, defining a mechanical system, where s is the distance from a fixed point
after t seconds, m is a mass, a the damping factor
and k the spring stiffness.
(iii)
d 2 y
dx 2 +
P
E I
y = 0, representing an equation for the
deflected profile y of a pin-ended uniform strut
of length l subjected to a load P. E is Young’s
modulus and I is the second moment of area.
If D represents
d
dx
and D 2 represents
d 2
dx 2 then the above
equation may be stated as
(aD 2 + bD + c)y = 0. This equation is said to be in
‘D-operator’ form.
If y = Ae mx then
d y
dx
= Ame mx and
d 2 y
dx 2 = Am 2 e mx .
Substituting these values into a
d 2 y
dx 2 + b
d y
dx
+ cy = 0
gives:
a(Am
2 e
mx
) + b(Ame
mx
) + c(Ae
mx
) = 0
i.e.
Ae
mx
(am
2
+ bm + c) = 0
Thus y = Ae mx is a solution of the given equation
provided that (am 2 + bm +c) = 0. am 2 + bm + c = 0 is
called the auxiliary equation, and since the equation is
a quadratic, m may be obtained either by factorizing or
by using the quadratic formula. Since, in the auxiliary
equation, a, b and c are real values, then the equation
may have either
(i) two different real roots (when b 2 > 4ac) or
(ii) two equal real roots (when b 2 = 4ac) or
(iii) two complex roots (when b 2 < 4ac).
Second order differential
equations of the form
a
d 2 y
dx 2 + b
dy
dx + cy = 0
50.1 Introduction
An equation of the form a
d 2 y
dx 2 + b
d y
dx
+ cy = 0, where
a, b and c are constants, is called a linear second
order differential equation with constant coefficients. When the right-hand side of the differential
equation is zero, it is referred to as a homogeneous
differential equation. When the right-hand side is not
equal to zero (as in Chapter 51) it is referred to as a
non-homogeneous differential equation.
There are numerous engineering examples of second
order differential equations. Three examples are:
(i) L
d 2 q
dt 2 + R
dq
dt
+
1
C
q = 0, representing an equation for charge q in an electrical circuit containing
resistance R, inductance L and capacitance C in
series.
(ii) m
d 2 s
dt 2 + a
ds
dt
+ ks = 0, defining a mechanical system, where s is the distance from a fixed point
after t seconds, m is a mass, a the damping factor
and k the spring stiffness.
(iii)
d 2 y
dx 2 +
P
E I
y = 0, representing an equation for the
deflected profile y of a pin-ended uniform strut
of length l subjected to a load P. E is Young’s
modulus and I is the second moment of area.
If D represents
d
dx
and D 2 represents
d 2
dx 2 then the above
equation may be stated as
(aD 2 + bD + c)y = 0. This equation is said to be in
‘D-operator’ form.
If y = Ae mx then
d y
dx
= Ame mx and
d 2 y
dx 2 = Am 2 e mx .
Substituting these values into a
d 2 y
dx 2 + b
d y
dx
+ cy = 0
gives:
a(Am
2 e
mx
) + b(Ame
mx
) + c(Ae
mx
) = 0
i.e.
Ae
mx
(am
2
+ bm + c) = 0
Thus y = Ae mx is a solution of the given equation
provided that (am 2 + bm +c) = 0. am 2 + bm + c = 0 is
called the auxiliary equation, and since the equation is
a quadratic, m may be obtained either by factorizing or
by using the quadratic formula. Since, in the auxiliary
equation, a, b and c are real values, then the equation
may have either
(i) two different real roots (when b 2 > 4ac) or
(ii) two equal real roots (when b 2 = 4ac) or
(iii) two complex roots (when b 2 < 4ac).