Second Order Differential Equations
Higher Order Linear Differential Equations
Ordinary Differential Equations
Dr. Marco A Roque Sol
12/01/2015
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Example 60
Solve the initial value problem and plot the solution.
u 00 + u = 0.5cos(0.8t),
u(0) = 0, u 0 (0) = 0
Solution
The general solution of is
u = c1 cos(ω0 t) + c2 sin(ω0 t) +
F0
cos(ωt)
− ω2)
m(ω02
Applying initial conditions, we obtain
c1 = −
F0
;
m(ω02 − ω 2 )
Dr. Marco A Roque Sol
c2 = 0
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
and the particuar solution is
u=
F0
(cos(ωt) − cos(ω0 t))
− ω2)
m(ω02
This is the sum of two periodic functions of different periods but
the same amplitude. Making use of the trigonometric identities for
cos(A ± B) with A = (ω0 + ω)t/2 and B = (ω0 − ω)t/2, we can
write the above equation in the form
F0
(ω0 − ω)t
(ω0 + ω)t
u=
sin
sin
2
2
m(ω02 − ω 2 )
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
If |ω0 − ω| is small, then ω0 + ω is much greater than it.
Consequently, sin(ω0 + ω)t/2 is a rapidly oscillating function
compared to sin(ω0 − ω)t/2. Thus the motion is a rapid oscillation
with frequency (ω0 + ω)/2 but with a slowly varying sinusoidal
amplitude
F0
sin (ω0 − ωt) 2
m|ω02 − ω 2 | This type of motion, possessing a periodic variation of amplitude,
exhibits what is called a beat. In this case ω0 = 1, = 0.8, and
F0 = 0.5, so the solution of the given problem is
u(t) = [2.77sin(0.1t)] sin(0.9t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Forced, Damped Vibrations
This is the full blown case where we consider every last possible
force that can act upon the system. The differential equation for
this case is,
mu 00 + γu 0 + ku = F (t)
The displacement function this time will be,
u(t) = uc + UP
where the complementary solution will be the solution to the ( free,
damped) homogeneous case and the particular solution will be
found using undetermined coefficients or variation of parameters.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
There are a couple of things to note here about this case. First,
from our work back in the free, damped case we know that the
complementary solution will approach zero as t → ∞. Because of
this, the complementary solution is often called the transient
solution in this case.
Also, because of this behavior the displacement will start to look
more and more like the particular solution as t increases and so the
particular solution is often called the steady state solution or
forced response.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Example 61
Take the system from the example 59 and add in a damper that
will exert a force of 45 Newtons when then velocity is 50 cm/sec.
Solution
So, all we need to do is compute the damping coefficient for this
problem then pull everything else down from the previous problem.
The damping coefficient is
Fd = γu 0 =⇒ 45 = γ(0.5) =⇒ γ = 90
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
The IVP for this problem is.
3u 00 + 90u 0 + 75u = 10cos(5t);
u(0) = 0.2, u 0 (0) = −0.1
The complementary solution for this example is
√
u(t) = c1 e (−15+10
2)t
√
+ c2 e (−15−10
2)t
For the particular solution we the form will be,
UP = Acos(5t) + Bsin(5t)
Plugging this into the differential equation and simplifying gives us,
405Bcos(5t) − 450Asin(5t) = 10cos(5t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Setting coefficient equal gives,
UP =
1
sin(5t)
45
The general solution is then
√
u(t) = c1 e (−15+10
2)t
√
+ c2 e (−15−10
2)t
+
1
sin(5t)
45
Applying the initial condition gives
√
u(t) = 0.1986e (−15+10
2)t
√
+ 0.0014e (−15−10
Dr. Marco A Roque Sol
2)t
+
Ordinary Differential Equations
1
sin(5t)
45
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Electric Circuits
A second example of the occurrence of second order linear
differential equations with constant coefficients is their use as a
model of the flow of electric current in the simple series circuit
shown below
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
The current I , measured in amperes (A), is a function of time t.
The resistance R in ohms (Ω), the capacitance C in farads (F ),
and the inductance L in henrys (H) are all positive and are
assumed to be known constants.
The impressed voltage E in volts (V ) is a given function of time.
Another physical quantity that enters the discussion is the total
charge Q in coulombs (C ) on the capacitor at time t. The relation
between charge Q and current I is
I =
Dr. Marco A Roque Sol
dQ
dt
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
The flow of current in the circuit is governed by Kirchhoffs second
law:
(Gustav Kirchhoff (18241887) was a German physicist and
professor at Breslau, Heidelberg, and Berlin. http://www.
britannica.com/biography/Gustav-Robert-Kirchhoff )
In a closed circuit the impressed voltage is equal to the sum of the
voltage drops in the rest of the circuit. According to the
elementary laws of electricity, we know that
The voltage drop across the resistor is IR.
The voltage drop across the capacitor is Q/C .
The voltage drop across the inductor is LdI /dt.
Hence, by Kirchhoffs law,
L
1
dI
+ RI + Q = E (t)
dt
C
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
The units for voltage, resistance, current, charge, capacitance,
inductance, and time are all related:
1volt = 1ohm × 1ampere = 1coulomb/1farad = 1henry × 1ampere/1seco
Substituting dQ
dt for I in the previous equation, we obtain the
differential equation
1
d 2Q
dQ
+ Q = E (t)
+R
2
dt
dt
C
for the charge Q. The initial conditions are
L
Q(t0 ) = Q0 ,
Q 0 (t0 ) = I (t0 ) = I0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Thus, we must know the charge on the capacitor and the current
in the circuit at some initial time t0 . Alternatively, we can obtain a
differential equation for the current I by differentiating the above
equation with respect to t, and then substituting dQ/dt for I . The
result is
dI
1
d 2I
+ R + I = E 0 (t)
2
dt
dt
C
with the initial conditions
L
I (t0 ) = I0 ,
I 0 (t0 ) = I00
it follows from the equation for Q(t) that
I00 =
E (t0 ) − RI0 − (1/C )Q0
L
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
The most important conclusion from this discussion is that the
flow of current in the circuit is described by an initial value
problem of precisely the same form as the one that describes the
motion of a springmass system.
m
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equation
An nth order linear differential equation, is an equation of the form
P0 (t)
d ny
d n−1 y
dy
+ Pn (t)y = G (t)
+
P
(t)
+ ... + Pn−1 (t)
1
n
n−1
dt
dt
dt
We assume that the functions P0 , ..., Pn , and G are continuous
real-valued functions on some interval I: α < t < β, and that P0 is
nowhere zero in this interval. Then, dividing the above equation by
P0 (t), we obtain
L[y ](t) =
d n−1 y
dy
d ny
+
p
(t)
+ ... + pn−1 (t)
+ pn (t)y = g (t)
1
n
n−1
dt
dt
dt
The linear differential operator L of order n defined above is similar
to the second order operator introduced before.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
The mathematical theory associated with is completely analogous
to that for the second order linear equation; for this reason we
simply state the results for the nth order problem.
Since the previous equation involves the nth derivative of y with
respect to t, we expect that to obtain a unique solution it is
necessary to specify n initial conditions
y (t0 ) = y0 ,
y 0 (t0 ) = y00 ,
(n−1)
..., y (n−1) (t0 ) = y0
(n−1)
where t0 may be any point in the interval I and y0 , y00 , ..., y0
any set of prescribed real constants
Dr. Marco A Roque Sol
Ordinary Differential Equations
, is
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
Theorem 4.1
If the functions p1 , p2 , ..., pn , and g are continuous on the open
interval I, then there exists exactly one solution y = φ(t) of the
differential equation that also satisfies the initial conditions
prescribed above, where t0 is any point in I. This solution exists
throughout the interval I.
The Homogeneus Equation
As in the corresponding second order problem, we first discuss the
homogeneous equation
L[y ](t) =
d ny
d n−1 y
dy
+ p1 (t) n−1 + ... + pn−1
+ pn (t)y = 0
n
dt
dt
dt
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
If the functions y1 , y2 , ..., yn are solutions of this equation, then it
follows by direct computation that the linear combination
y (t) = c1 y1 (t) + c2 y2 (t) + ... + cn yn (t)
where c1 , c2 , ..., cn are arbitrary constants, is also a solution.
Can every solution of homogeneus equation be expressed as a
linear combination of y1 , ..., yn ? This will be true if, it is possible
to choose the constants c1 , ..., cn so that the linear combination
given above satisfies the initial conditions. That is, for any choice
(n−1)
of the point t0 in I, and for any choice of y0 , y00 , ..., y0
,
(n−1)
assuming that y (t) belongs to CI =[α,β] , we must be able to
determine c1 , ..., cn so that the equations
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
c1 y1 (t0 ) + c2 y2 (t0 ) + ... + cn yn (t0 ) = y0
c1 y10 (t0 ) + c2 y20 (t0 ) + ... + cn yn0 (t0 ) = y00
..
.
(n−1)
c1 y1
(n−1)
(t0 ) + c2 y2
(n−1)
(t0 ) + ... + cn yn
(t0 ) = y0
are satisfied. This system can be solved uniquely for the constants
c1 , ..., cn , provided that the determinant of coefficients is not zero.
On the other hand, if the determinant of coefficients is zero, then
(n1)
it is always possible to choose values of y0 , y00 , ..., y0
so that the
equations do not have a solution.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
Therefore a necessary and sufficient condition for the existence of a
(n−1)
solution of above equations for arbitrary values of y0 , y00 , ..., y0
is that the Wronskian
y1 (t0 )
y
(t
)
.
.
.
y
2
0
n
0
0
0
y1 (t0 )
y2 (t0 )
...
yn W =
..
..
..
.
.
.
(n−1)
(n−1)
(n−1) y
(t0 ) y2
(t0 ) . . . yn
1
is not zero at t = t0 .
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
Theorem 4.2
If the functions p1 , p2 , ..., pn are continuous on the open interval I,
if the functions y1 , y2 , ..., yn are solutions of the homogeneous
equation, and if W (y1 , y2 , ..., yn )(t) 6= 0 for at least one point in I,
then every solution can be expressed as a linear combination of the
solutions y1 , y2 , ..., yn .
A set of solutions y1 , ..., yn of the homogeneus equation
L[y ](t) =
d ny
d n−1 y
dy
+
p
(t)
+ ... + pn−1
+ pn (t)y = 0
1
dt n
dt n−1
dt
whose Wronskian is nonzero is referred to as a fundamental set
of solutions. Since all solutions of this equation are of the form
y (t) = c1 y1 (t) + c2 y2 (t) + ... + cn yn (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
we use the term general solution to refer to an arbitrary linear
combination of any fundamental set of solutions of homogeneous
equation.
Linear Dependence and Independence .
We now explore the relationship between fundamental sets of
solutions and the concept of linear independence, a central idea in
the study of linear algebra ( Intutively, Linear Algebra, is the study
of Linear transformations between Vector Spaces ). The functions
f1 , f2 , ..., fn are said to be linearly dependent on an interval I if
there exists a set of constants k1 , k2 , ..., kn ,not all zero , such that
k1 f1 (t) + k2 f2 (t) + ... + kn fn (t) = 0
for all t in I. Otherwise, the functions f1 , ..., fn are said to be
linearly independent on I .
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
General Theory of nth Order Linear Equations
Example 4.1
Determine whether the functions
f1 (t) = 1, f2 (t) = t, and f3 (t) = t2
are linearly independent or dependent on the interval
I : −∞ < t < ∞.
Solution
Form the linear combination
k1 f1 (t) + k2 f2 (t) + k3 f3 (t) = k1 (1) + k2 (t) + k3 (t 2 ) = 0
And differentiating the above formula twice
k2 + 2k3 t = 0
2k3 = 0
Therefore k1 = k2 = k3 = 0 and the set of functions is linearly
independent.
Dr. Marco A Roque Sol
Ordinary Differential Equations