Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Ordinary Differential Equations
Dr. Marco A Roque Sol
12/01/2015
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
The Method of Variation of Parameters for determining a
particular solution of the nonhomogeneous equation, YP (t), rests
on the possibility of determining n functions u1 , u2 , ..., un such that
YP (t) is of the form
YP (t) = u1 (t)y1 (t) + u2 (t)y2 (t) + ... + un (t)yn (t)
Since we have n functions to determine, we will have to specify n
conditions. One of these is clearly that YP the ODE. The other
n − 1 conditions are chosen arbitrarily. as to make the calculations
as simple as possible. Following the same idea used in the second
order case we have that the first derivative of YP is given by
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
YP0 (t) = u1 (t)y10 (t) + u2 (t)y20 (t) + ... + un (t)yn0 (t) +
u10 (t)y1 (t) + u20 (t)y2 (t) + ... + un0 (t)yn (t)
Thus the first condition that we impose is that
u10 (t)y1 (t) + u20 (t)y2 (t) + ... + un0 (t)yn (t) = 0
We continue this process by calculating the successive derivatives
Y 00 , ..., Y (n−1) . After each differentiation we set equal to zero the
sum of terms involving derivatives of u1 , ..., un . In this way we
obtain n − 2 further conditions similar to the above and put them
together we have
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
(m)
(m)
(m)
u10 (t)y1 (t) + u20 (t)y2 (t) + ... + un0 (t)yn (t) = 0;
m = 1, 2, ..., n − 1
As a result of these conditions, it follows that the expressions for
(n−1)
YP0 , YP00 , ..., YP
reduce to
(m)
YP
(m)
(m)
(m)
= u1 (t)y1 (t) + u2 (t)y2 (t) + ... + un (t)yn (t) = 0;
m = 1, 2, 3, ..., n − 1
Finally, we need to impose the condition that YP must be a
(n1)
solution of the nonhomogeneous equation. By differentiating YP
from the above equation, we obtain
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
(n)
YP
(n)
(n)
(n)
) = u1 (t)y1 (t) + u2 (t)y2 (t) + ... + un (t)yn (t) +
(n−1) 0
= (YP
(n−1)
u10 (t)y1
(n−1)
(t) + u20 (t)y2
(n−1)
(t) + ... + un0 (t)yn
(t) = g (t)
plug into the equation and considering that L[yi ] = 0; i = 1, 2, ..., n,
then the ramaining terms yield the relation
(n−1)
u10 (t)y1
(n−1)
(t) + u20 (t)y2
(n−1)
(t) + ... + un0 (t)yn
(t) = g (t)
Thus, we have a system of n simultaneos linear nonhomogeneus
algebraic equations for u10 , u20 , ..., un0 :
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
u10 (t)y1 (t) + u20 (t)y2 (t) + ... + un0 (t)yn (t) = 0
u10 (t)y10 (t) + u20 (t)y20 (t) + ... + un0 (t)yn0 (t) = 0
u10 (t)y100 (t) + u20 (t)y200 (t) + ... + un0 (t)yn00 (t) = 0
..
.
(n−1)
u10 (t)y1
(n−1)
(t) + u20 (t)y2
(n−1)
(t) + ... + un0 (t)yn
Dr. Marco A Roque Sol
(t) = g (t)
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters







y1
y10
y100
y2
y20
y200
(n−1)
y1
(n−1)
y2
...
...
...
..
.
yn
yn0
yn00
(n−1)
... yn
 0 
u1
 u 0  
  2 
 u 0  
  3 = 
  ..  
 .  
un0
0
0
0
..
.







g (t)
The above system, is a linear algebraic system for the unknown
quantities u10 , u20 , ..., un0 . The determinant of coefficients is precisely
W (y1 , y2 , ..., yn ), and it is nowhere zero since y1 , ..., yn is a
fundamental set of solutions of the homogeneous equation. Hence
it is possible to determine u10 , u20 , ..., un0 using Cramer’s Rule
(http://www.purplemath.com/modules/cramers.htm):
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
0
um
(t) =
g (t)Wm (t)
;
W (t)
m = 1, 2, ..., n
where W (t) = W (y1 , y2 , ..., yn )(t) ( The Wronskian ) and Wm (t)
is the determinant obtained from W by replacing the mth column
by the column (0, 0, ..., 0, 1).
y1
y2
...
yn 0
y1
y20
...
yn0 00
00
y2
...
yn00 W = y1
..
.
(n−1)
(n−1)
(n−1) y
y
... y
1
Dr. Marco A Roque Sol
2
n
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
m − th
y1
...
0
0
y1
...
0
Wm = y 00
....
0
1
..
.
(n−1)
y
...
1
1
...
...
...
...
yn yn0 yn00 (n−1) y
n
And integrating the above equations we have that the particular
solution is given by
YP (t) =
n
X
Z
t
ym
m=1
Dr. Marco A Roque Sol
t0
g (s)Wm
ds
W (s)
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
where t0 is an arbitrary point. As we can see, things get more
complicated than in the second order case. In some cases the
calculations may be simplified to some extent by using Abel’s
identity
Z
W (t) = W (y1 , y2 , ..., yn )(t) = cexp − p1 (t)dt
The constant c can be determined by evaluating W at some
convenient point.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
Example 4.7
Given that y1 (t) = e t , y2 (t) = te t , and y3 (t) = e −t are solutions
of the homogeneous equation corresponding to
y 000 − y 00 − y 0 + y = g (t)
determine a particular in terms of an integral.
Solution
Let’s determine first the Wronskian
t
e
t
t −t
W = W (e , te , e )(t) = e t
e t
Dr. Marco A Roque Sol
te t
(t + 1)e t
(t + 2)e t
e −t −e −t e −t Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
Factoring e t from each of the first two columns pause and e −t
from the third column, we obtain
1
t
1 W = W (e t , te t , e −t )(t) = e t 1 (t + 1) −1
1 (t + 2) 1 Then, by subtracting the first row from the second and third rows,
we have
1 t 1 W = W (e t , te t , e −t )(t) = e t 0 1 −2
0 2 0 Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
The Method of Variation of Parameters
The Method of Variation of Parameters
Finally, evaluating this determinant by minors using the first
column, we find that
W (t) = 4e t ;
W1 (t) = −2t − 1;
W2 (t) = 2;
W3 (t) = e 2t
Thus, the particular solution is given by
Z t
3
X
g (s)Wm
YP (t) =
ym
t0 W (s)
m=1
YP (t) = e t
Z
t
t0
g (s)(−1 − 2s)
+ te t
4e s
1
YP (t) =
4
Z
t
t0
g (s)(2)
+ e −t
4e s
Z
t
t0
g (s)e 2s
+
4e s
Z th
i
e t−s (−1 + 2(t − s)) + e −(t−s) g (s)ds
t0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
1.− A power series
∞
X
an (x − x0 )n is said to be convergent at a
n=0
point x if
limm→∞
Sm =
m
X
!
an (x − x0 )n
n=0
exists for that x. The series certainly converges for x = x0 ; it may
converge for all x, or it may converge for some values of x and not
for others.
2. The series
∞
X
an (x − x0 )n is said to converge absolutely at a
n=0
point x if the series
m
X
|an (x − x0 )n |
n=0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
converges. It can be shown that if the series converges absolutely,
then the series also converges; however, the converse is not
necessarily true.
3. One of the most useful tests for the absolute convergence of a
power series is the ratio test. If an 6= 0, and if, for a fixed value of
x,
an+1 an+1 (x − x0 )n+1 = |x − x0 |L
limn→∞ = |x − x0 |limn→∞ an (x − x0 )n an then the power series converges absolutely at that value of x if
|x − x0 |L < 1 and diverges if |x − x0 |L > 1. If |x − x0 |L = 1, the
test is inconclusive.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
P
n
4. If the power series, ∞
n=0 an (x − x0 ) converges at x = x1 , it
converges absolutely for |x − x0 | < |x1 − x0 |; and if it diverges at
x = x1 , it diverges for |x − x0 | > |x1 − x0 |
5. For a typical power series, there is a P
positive number ρ, called
n
the radius of convergence, such that ∞
n=0 an (x − x0 )
converges absolutely for |x − x0 | < ρ and diverges for |x − x0 | > ρ.
The interval |x − x0 | < ρ is called the interval of convergence
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
P
P∞
n
n
Suppose that, ∞
n=0 an (x − x0 ) and
n=0 bn (x − x0 ) converge
to f (x) and g (x), respectively, for |x − x0| < ρ, ρ > 0.
6. The two series can be added or subtracted termwise, and
f (x) ± g (x) =
∞
X
an (x − x0 )n ± bn (x − x0 )n
n=0
the resulting series converges at least for |x − x0 | < ρ
7. The two series can be formally multiplied
f (x)g (x) =
"∞
X
#
n
an (x − x0 )
n=0
[bn (x − x0 )n ] =
∞
X
cn (x − x0 )n
n=0
where cn = a0 bn + a1 bn−1 + a2 bn−2 + ... + an b0 . The resulting
series converges at least for |x − x0| < ρ.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
Moreover, if g (x0 ) 6= 0, the series for f (x) can be formally divided
by the series for g (x), and
∞
f (x) X
=
dn (x − x0 )n
g (x)
n=0
where dn can be found as
f (x) =
∞
X
"
an (x − x0 )n =
n=0
∞
X
#
dn (x − x0 )n [bn (x − x0 )n ] =
n=0
=
∞
n
X
X
n=0
!
dk bn−k
(x − x0 )n
k=0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
The resulting power series, may have a radius of convergence less
than ρ.
8. The function f is continuous and has derivatives of all orders for
|x − x0 | < ρ. Moreover, f 0 , f 00 , ... can be computed by
differentiating the series termwise, that is,
f 0 (x) = a1 (x − x0 ) + a2 (x − x0 )2 + ... + nan (x − x0 )n−1 + .... =
∞
X
nan (x − x0 )n−1
n=1
00
f (x) = 2a2 + 6a3 (x − x0 ) + ... + n(n − 1)an (x − x0 )n−2 + .... =
∞
X
n(n − 1)an (x − x0 )n−2
n=2
and so forth, and each of the series converges absolutely for
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
9. If the value of an is given by
f (n) (x0 )
n!
The series is called the Taylor series for the function f about
x = x0 .
an =
P
P∞
n
n
10. If ∞
n=0 an (x − x0 ) =
n=0 bn (x − x0 ) for each x in some
open intervalP
with center x0 , then an = bn for n = 0, 1, 2, 3, ... . In
n
particular, if ∞
n=0 an (x − x0 ) = 0 for each x, then
a0 = a1 = ... = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
A function f that has a Taylor series expansion about x = x0 can
be written as
f (x) =
∞
X
f (n) (x0 )
n=0
n!
(x − x0 )n
Example 5.1
For which values of x does the power series
∞
X
(−1)n+1 n(x − 2)n
n=0
converges ?
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
Solution
We use the ratio test
n + 1
(−1)n+2 (n + 1)(x − 2)n+1 limn→∞ = |x − 2|limn→∞ n = |x − 2|
(−1)n+1 n(x − 2)n
Thus, the series converges absolutely for |x − 2| < 1, or 1 < x < 3,
and diverges for |x − 2| > 1. The test is inconclusive if |x − 2| = 1,
that is, x = 1 and x = 3 for each of these values of x. In
particular, for those values limn→∞ an 6= o and therefore the series
is divergent.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
Example 5.2
Determine the radius of convergence of the power series
∞
X
(x + 1)n
n2n
n=1
converges ?
Solution
We use the ratio test
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Review of Power Series
(x + 1)n+1 /(n + 1)2n+1 |x − 2|
n |x + 1|
=
limn→∞ limn→∞ =
(x + 1)n /n2n
s
n + 1
2
Thus the series converges absolutely for |x + 1| < 2 , or
−3 < x < 1, and diverges for |x + 1| > 2. The radius of
convergence is ρ = 2. At the end points we get for: x = 1
∞
X
1
n=1
n
a divergent series. At x = −3
∞
X
(−1)n
n=1
n
which is a (conditionally) convergent alternating series. Therefore,
the series is absolutely convergent for −3 < x < 1
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Series Solutions Near an Ordinary Point, Part I
We will consider methods of solving second order linear equations
when the coefficients are not constants. It is sufficient to consider
the homogeneous equation
P(x)
dy
d 2y
+ Q(x)
+ R(x) = 0
dx 2
dx
To simplify the problems we will be dealing only with polynomial
coefficients (continuous functions).
Many problems in mathematical physics lead to this equations;
examples include the Bessel equation
(https:en.wikipedia.org/wiki/Friedrich_Bessel )
x 2 y 00 + xy 0 + (x 2 − ν 2 )y = 0
where ν is a constant.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Series Solutions Near an Ordinary Point, Part I
Bessel equation can be found in:
1) Electromagnetic waves in a cylindrical waveguide;
2) Heat conduction in a cylindrical object;
3) Modes of vibration of a thin circular (or annular) acoustic
membrane (such as a drum or other membranophone
https://en.wikipedia.org/wiki/Vibrations_of_a_
circular_membrane );
4) Diffusion problems on a lattice;
5) Solutions to the radial Schrdinger equation (in spherical and
cylindrical coordinates) for a free particle.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
Series Solutions of Second Order Linear Equations
Review of Power Series
Series Solutions Near an Ordinary Point, Part I
Series Solutions Near an Ordinary Point, Part I
And the Legendre equation
(https://en.wikipedia.org/wiki/Adrien-Marie_Legendre )
(1 − x 2 )y 00 − 2xy 0 + α(α + 1)y = 0
where α is a constant.
The Legendre equation can be found in:
1)
2)
3)
4)
Quantum mechanical model of the hydrogen atom;
Solving Laplaces equation in spherical coodrinates;
Steady state temperature within a solid spherical ball;
The gravitational potential associated to a point mass.
Dr. Marco A Roque Sol
Ordinary Differential Equations