Second Order 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
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Nonhomogeneous Equations; Method of
Undetermined Coefficients
b) y 00 − 100y = 9t 2 e 10t + cos(t) + tsin(t)
The general solution of the associate homogeneus equation
(y 00 − 100y = 0 =⇒ r 2 − 100 = (r − 10)(r + 10) = 0
=⇒ r1 = 10, r2 = −10) is
y (t) = c1 e 10t + c2 e −10t
Thus, the form of the particular solution is
YP = (At 2 + Bt + C )e 10t t + (Et + F )cos(t) + (Gt + H)sin(t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Nonhomogeneous Equations; Method of
Undetermined Coefficients
c)
4y 00 + y = e −2t sin(t/2) + 6tcos(t/2)
The general solution of the associate homogeneus equation
(4y 00 + y = 0 =⇒ 4r 2 + 1 = (r − i/2)(r + i/2) = 0 =⇒
r1 = i/2, r2 = −i/2) is
y (t) = c1 cos(t/2) + c2 sin(t/2)
Thus, the form of the particular solution is
YP = e −2t (Acos(t/2) + Bsin(t/2)) + (Ct + D)cos(t/2)t + ...
.... + (Et + F )sin(t/2)t
YP = (Acos(t/2) + Bsin(t/2)) e −2t + ...
... + t(Ct + D)cos(t/2) + t(Et + F )sin(t/2)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Nonhomogeneous Equations; Method of
Undetermined Coefficients
d)
4y 00 + 16y 0 + 17y = e −2t sin(t/2) + 6tcos(t/2)
The general solution of the associate homogeneus equation
(4y 00 + 16y 0 + 17y = 0 =⇒ 4r 2 + 16r + 17 = 0 =⇒
r1 = −2 + i/2, r2 = −2 − i/2) is
y (t) = c1 e −2t cos(t/2) + c2 e −2t sin(t/2)
Thus, the form of the particular solution is
YP = (Acos(t/2) + Bsin(t/2)) e −2t t + ...
... + (Ct + D)cos(t/2) + (Et + F )sin(t/2)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Nonhomogeneous Equations; Method of
Undetermined Coefficients
e) y 00 + 8y 0 + 16y = e −4t + (t 2 + 5)e −4t = e −4t (t 2 + 6)
The general solution of the associate homogeneus equation
(y 00 + 8y 0 + 16y = 0 =⇒ r 2 + 8r + 16 = (r − 4)2 = 0
=⇒ r1 = r2 = −4) is
y (t) = c1 e −4t + c2 te −4t
Thus, the form of the particular solution is
YP = At 2 + Bt + C e −4t t
but,
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Nonhomogeneous Equations; Method of
Undetermined Coefficients
still the term te −4t is a solution of the associate homogeneous
equation !!!!! Therefore the final version of YP is
YP = At 2 + Bt + C e −4t t 2
or
YP = t 2 At 2 + Bt + C e −4t
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Variation of Parameters
In this section we describe another method of finding a particular
solution of a nonhomogeneous equation.
y 00 + p(t)y 0 + q(t)y = g (t)
This method,variation of parameters, is due to Lagrange (
http://www-gap.dcs.st-and.ac.uk/history/Biographies/
Lagrange.html ) . The main advantage of variation of
parameters is that it is a general method. However, there are two
disadvantages to the method. First, the solution of the associate
homogeneus equation is absolutely required. Second, as we will
see, in order to complete the method we will be doing a couple of
integrals and there is no guarantee that we will be able to do them.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Variation of Parameters
Let’s start deriving he formula for variation of parameters. We’ll
start off by taking into account that the solution to
y 00 + p(t)y 0 + q(t)y = 0
is of the form
y (t) = c1 y1 + c2 y2
The crucial idea is to replace the constants c1 and c2 in the above
equation by functions u1 (t) and u2 (t), respectively, thus we have
YP (t) = u1 y1 + u2 y2
and from there, we obtain
YP0 (t) = u1 y10 + u10 y1 + u2 y20 + u20 y2
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Variation of Parameters
Now, we have two unknowns functions u1 and u2 but only one
equation to use, so we impose an extra condition to determine
these two functions. Namely, in the equation for YP0 we set the
sum of the terms involving u10 and u20 equal to zero
u10 y1 + u20 y2 = 0
Therefore, YP0 (t) = u1 y10 + u2 y20 , and
YP00 = u10 y10 + u1 y100 + u20 y20 + u2 y200
Now, we substitute for y , y 0 , and y 00 in the ODE and after
rearranging the terms in the resulting equation, we find that
YP00 + p(t)YP0 + q(t)YP = g (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Variation of Parameters
(u10 y10 + u1 y100 + u20 y20 + u2 y200 )00 + p(t)(u1 y10 + u2 y20 )0 + ...
... + q(t)(u1 y1 + u2 y2 ) = g (t)
u1 (y100 + p(t)y1 + q(t)y1 ) + u2 (y200 + p(t)y2 + q(t)y2 ) + ...
... + (u10 y10 + u20 y20 ) = g (t) + ...
The first two expressions in the parenthesis are zero because both
y1 and y2 are solutions of the homogeneous equation. Thererefore,
we get
u10 y10 + u20 y20 = g (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Variation of Parameters
Thus, the system of equations for the two unknowns is
u10 y1 + u2 y20 = 0
u10 y10 + u20 y20 = g (t)
whose solution is given by
u10 = −
y2 g (t)
;
W (y1 , y2 )(t)
u20 =
y1 g (t)
W (y1 , y2 )(t)
where W (y1 , y2 ) is the Wronskian of y1 and y2 . By integrating the
above equations, we find the desired functions u1 (t) and u2 (t),
namely,
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Variation of Parameters
Z
u1 = −
y2 g (t)
dt + c1 ;
W (y1 , y2 )(t)
Z
u2 =
y1 g (t)
dt + c2
W (y1 , y2 )(t)
Therorem
If the functions p, q, and g are continuous on an open interval I,
and if the functions y1 and y2 are a fundamental set of solutions of
the homogeneous equation corresponding to the nonhomogeneous
equation
y 00 + p(t)y 0 + q(t)y = g (t)
then a particular solution is
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Variation of Parameters
Z
1
YP = −y1 (t)
t0
y2 (s)g (s)
ds + y2 (t)
W (y1 , y2 )(s)
Z
t
t0
y1 (s)g (s)
ds
W (y1 , y2 )(s)
where t0 is any conveniently chosen point in I. The general solution
is
y = c1 y1 (t) + c2 y2 (t) + YP (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Second order linear equations with constant coefficients are
important in two physical processes, namely, Mechanical and
Electrical oscillations.
Actually from the Math point of view, both problems are the same.
However, from the Physics point of view thay are quite different.
For example, the motion of a mass on a vibrating spring, the
angular motion of a simple pendulum, the flow of electric current
in a simple series circuit and the electrical charge in an electric
circuit, are just examples of that difference.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Let’s get the situation setup. We are going to start with a spring
of length l, called the natural length, and we’re going to hook an
object with mass m up to it. When the object is attached to the
spring the spring will stretch a length of L. Below is sketch of the
spring with and without the object attached to it.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Convention
As denoted in the sketch we are going to assume that all forces,
velocities, and displacements in the downward direction will be
positive. All forces, velocities, and displacements in the upward
direction will be negative.
Also, as shown in the sketch above, we will measure all
displacement of the mass from its equilibrium position. Therefore,
the u = 0 position will correspond to the center of gravity for the
mass as it hangs on the spring and is at rest.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Now, we need to develop a differential equation that will give the
displacement of the object at any time t. First, recall Newton’s
Second Law of Motion.
F = ma
In this case we will use the second derivative of the displacement,
u, for the acceleration and so Newton’s Second Law becomes,
F (t, u, u 0 ) = mu 00
Here is a list of the forces that will act upon the object.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Gravity, Fg
The force due to gravity will always act upon the object of course.
This force is
Fg = mg
Spring, Fs
We are going to assume that Hookes Law will govern the force
that the spring exerts on the object. This force will always be
present as well and is
Fs = −k(L + u)
Hookes Law tells us that the force exerted by a spring will be the
spring constant, k > 0, times the displacement of the spring from
its natural length.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Damping, Fd
The next force that we need to consider is damping. This force
may or may not be present for any given problem. This force works
to counteract any movement. This damping force is
Fd = −γu
where, γ > 0 is the damping coefficient.
External Forces, F (t)
If there are any other forces acting on our object we collect them
in this term. We typically call F (t) the forcing function.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Putting all of these together gives us the following for Newtons
Second Law.
mu 00 = mg − k(L + u) − γu 0 + F (t)
Or, upon rewriting, we get,
mu 00 + γu 0 + ku = mg − kL + F (t)
Now, when the object is at rest in its equilibrium position,
mg − kL = 0
Using this in Newtons Second Law /pause gives us the final version
of the differential equation that well work with.
mu 00 + γu 0 + ku = F (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Second Order Differential Equations
Mechanical and Electrical Vibrations
Along with this differential equation we will have the following
initial conditions.
u(0) = u0
Initial
u 0 (0) = u00
displacement
Initial
velocity
OBS
If we have a mass m attached to a spring with constant k in a
surface with friction c ( or γ) and subject to an external force F (t),
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
then it satisfy the differential equation
mu 00 + cu 0 + ku = F (t)
Free, Undamped Vibrations
This is the simplest case that we can consider. Free or unforced
vibrations means that F (t) = 0 and undamped vibrations means
that γ = 0. In this case the differential equation becomes,
mu 00 + ku = 0
The characteristic equation has the roots,
r
r
k
k
r =±
= ±ω0 i; ω0 =
m
m
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Nonhomogeneous Equations; Method of Undetermined Coefficie
Variation of Parameters
Mechanical and Electrical Vibrations
Mechanical and Electrical Vibrations
Where ω0 is called the natural frequency. Recall as well that
m > 0 and k > 0 and so we can guarantee that this quantity will
not be complex. The solution in this case is then
u(t) = c1 cos(ω0 t) + c2 sin(ω0 t)
We can write the above equation in the following form
u(t) = Rcos(ω0 t − δ)
(If c1 = Rcos(δ) c2 = Rsin(δ) =⇒ u(t) = Rcos(δ)cos(ω0 t)+
Rsin(δ)sin(ω0 t) =⇒ u(t) = Rcos(ω0 t − δ); R 2 = c12 + c22 ;
tan(δ) = c2 /c1 )
where R is the amplitude of the displacement and δ is the
p mphase
shift or phase angle of the displacement. T = 2π
=
2π
ω0
k is
called the natural period.
Dr. Marco A Roque Sol
Ordinary Differential Equations