ABSTRACT
Boundary problems of ordinary dierential
equations
Linear dierential equations
Let us rst consider equations of second order According to chapter equa
tions of motion and other models combine the acceleration xt of a phe
nomenon with some external inuence like forces The most general linear
dierential equation of second order apparently has the form
p
xy
p
xy
p
xy fx
Here p
p
p
and f are in most cases given notvanishing continuous func
tions If fx is zero the equation is called homogeneous We now rst
solve the homogeneous equation
p
xy
p
xy
p
xy
Let
yx C
y
x C
y
x
be the general solution where y
and y
are fundamental solutions and C
and C
are constants of integration In the next section we will discuss meth
ods how one may nd y
and y
Let y
x be a particular solution of the
inhomogeneous equation then its general solution has the form
yx C
y
x C
y
x y
x
The next step is to nd a particular solution y
x of We do this by
replacing the constants C
and C
by nonconstant functions C
x and C
x
This method is called the method of variation of parameters of constants
because the constants in are now allowed to vary Instead of the
unknown function yx we now have two new functions additionally The
setup
y
x C
xy
x C
xy
x
delivers y
x and y
x Since the new functions C
x and C
x are quite
arbitrary we may require two new conditions
From and we then obtain
y
C
y
C
y
and from and one gets
y
C
y
C
y
fxp
x
Since is assumed to be a solution of insertion of
and into demonstrates that is actually a solution of
Now we determine the two stillunknown functions C
x and C
x
from and We obtain
C
x
y
xfx
p
x y
xy
x y
xy
x
C
x
y
xfx
p
x y
xy
x y
xy
x
Integrations yield
C
x
x
Z
x
y
f
p
W
d C
x
x
Z
x
y
f
p
W
d
The denominator appearing in is called the Wronskian determinant
W
W x y
xy
x y
xy
x
y
x y
x
y
x y
x
If this determinant vanishes then the two solutions y
x and y
x are lin
early dependent
C
xy
C
xy
for C
C
If one knows two independent solutions y
x and y
x
then also the particular solution y
is known
y
x
x
Z
x
y
y
x y
xy
W
f
p
d
If one solution y
x of is known then the second solution of
may be found
y
x y
x
Z
exp
R
p
xp
xdx
dx
yx zx exp
Z
p
x
p
x
dx
may transform away the y
x term in Inserting into
one obtains
p
xz
p
x
p
x
p
x
z fx exp
Z
p
x
p
x
dx
and neither y
nor z
appears The homogeneous equation now takes
the form
z
x Ixzx
The invariant
Ix
p
x
p
x
p
x
p
x
is a means to classify dierential equations of second order Thus the general
solutions of two dierential equations having the same invariant dier only by
a factor
The method just described can be applied on two examples We consider
the inhomogeneous equation
y
y
y expx
Its solution is given by
y C
expx C
expx expx
where C
and C
are constants The same result can be obtained by the
Mathematica command
DSolve[y [x]-3*y[x]+2*y[x]==Exp[5*x],y[x],x] We now consider the boundary problem twopoint problem
y y
of the equation
y
y
y expx
The general solution of is given by
y C
expx C
expx expx
Now the integration constants C
and C
can be determined by inserting
into Again the solution can be obtained by the
Mathematica command
Problems
For constant C
C
insert into and obtain the resulting
dierential equations for y
y
y
Does the solution of
satisfy these equations derived by you Answer should be yes Try to
use Mathematica for this calculation
Calculate the invariant I for the two equations and
Answer and Try to reproduce the solutions
and by using and
In Mathematica the Wronskian can be dened by a determi
nant Since a determinant is an operation on a matrix we rst have
to dene a matrix We use the solutions y
x and y
x contained in
M={{Exp[x],Exp[2*x]},{Exp[x],2*Exp[2*x]}} then Det[M] results in ex Calculate the Wronskian for the solution The answer should read e
x
Calculate the Wronskian for
a sinx cosx Answer cos
x sin
x
b sinx sinx Answer Why
c x
x
Answer x
Solve p
y
x p
y
x p
y for constant p
p
p
In order to
delete previous denitions for y we use
Clear[y];DSolve[p0*y[x]+p1*y[x]+p2*y[x]==0,y[x],x] The result looks complicated Simplify[%] does not help very much But is the result correct Can we verify the output of the calculation by
inserting it into the dierential equation To do so we bring the result
into the input form by using a new function ux The new function
must have x as an independent variable guaranteeing that ux is a
global function giving values for any x
The following example will clear the situation
u[x]=4*xˆ2 gives u[2]=u[2] but v[x_]:=4*xˆ2 gives v[2]=16 To verify the solution of the dierential equation we use again
solution
y xexp
p
p
p
p p x
p
C
exp
p
p
p
p p x
p
C
Solving linear dierential equations
As a rst example of the solution of a boundary value problem we consider
the linear dierential equation
y
y
which has the general solution
yx A sinxB cosx
A sinx or cosx would be particular solutions The general solution admits
both initial or boundary value problems If we choose the initial conditions
onepoint conditions
y
y
then the integration constants A and B can be obtained from
A sin
B cos
A
A cos
B sin
B
On the other hand if we choose the boundary conditions two point conditions
yx
y
y
x
y
we obtain from
A
cosx
cosx
B
sinx
sinx
sinx
cosx
sinx
cosx
Thus the general solution of the boundary value problem and
is given by the sum of two particular solutions
yx
cosx
cosx
sinx
cosx
sinx
cosx
sinx
sinx
sinx
sinx
cosx
sinx
cosx
cosx
A warning is now necessary not all arbitrary boundary problems can be
solved If we assume that
y
y
then the general solution is not able to satisfy these equations From
one obtains the contradiction
A and A
If we replace by
y y
we get an innity of solutions since B but A remains undetermined
The nonvanishing boundary conditions and are
called inhomogeneous Adversely the vanishing conditions
yx
yx
are called homogeneous
We now have the same situation that we discussed in section for partial
dierential equations A boundary problem is called homogeneous if the dif
ferential equation and the boundary conditions are both homogeneous If the
dierential equation or the boundary condition or both are inhomogeneous
then the boundary problem is said to be inhomogeneous
Having solved the homogeneous equation we now consider the in
homogeneous equation of oscillations
y
y fx
A particular solution of the homogeneous equation is given by yx A cosx
In order to solve the inhomogeneous equation we use the method of variation
of constants In analogy to we write
geneous equation for Ax
A
cosx A
sinx fx
Application of the Mathematica command
A x C
Z
x
K
C Sec K
Z
K
K
Cos K f K dK
Sec K
dK
This apparently means that Mathematica cant solve To solve the
equation step by step we consider the corresponding homogeneous equation
A
A
tanx
The substitution ux A
x u
x A
x gives the separable equation
du
u
tanxdx
Integration yields
A
x ux C exp
Z
tanxdx
C cos
x
It seems that Mathematica is not able to integrate equation Since
we do not need Ax itself we can now solve the inhomogeneous equation
directly by inserting A
x Cxcos
x into it The result after
some short algebra is
dC
dx
C
x fx cosx
and
Cx
Z
fx cosxdx
This result may also be derived with the help of Mathematica
f x Sec x C
x
Finally we obtain
Ax
Z
uxdx
Z
cos
x
Z
fx cosxdx
dx
and
yx
Z
Z
fx cosxdx
cos
xdx cosxA cosxB sinx
Here the rst term is the particular solution of the inhomogeneous equation
and the other two terms represent the general solution of the homo
geneous equation Solution is a consequence of the theorem that the
solution of an inhomogeneous linear equation consists of the superposition of
a particular solution of the inhomogeneous equation and the general solution
of the homogeneous equation
For the special function fx A sinx D where A D and are given
constants we now solve the boundary problem
yx
y
yx
y
Here y
and y
are given constant values With the function fx given the
solution takes two forms For resonance between the eigenfrequency
and the exterior excitation frequency ie for the solution is
yx
C
x sinxB sinx
where CB and are constant Solutions of this type are not able to satisfy
They are called secular and play a role in approximation theory
The second form of the solution is valid for and reads
yx
A
sinx
D
B sinx
If one combines this solution with the boundary conditions one gets
y
A
sinx
D
B sinx
y
A
sinx
D
B sinx
These equations determine the integration constants B and
Since we now know that inhomogeneous problems of linear equations can be
reduced to a homogeneous problem we restrict ourselves to discuss methods
Here the functions p
and p
are the functions p
p
and p
p
from
renamed The dierential equation is called to be of the Fuchsian
type if the functions are regular rational with exception of poles local regular
singular points To make this clear we consider the Euler equation
ax x
yx bx x
y
x cyx
which is a special case of and where a b and c are constants A
point x
is called an ordinary point if ax x
and a singular point if
axx
Near an ordinary point solutions of can be found using
the method of power series
P
n
a
n
x
n
Near a singular point the Frobenius
method will be used
Instead of we can consider Now bx x
ax x
will
be replaced by p
x Thus if ax x
then p
x singular point
pole A function regular everywhere but with one pole at x
can no longer
be expanded into a power series but it can be represented by a Laurent
series
P
n
a
n
x x
n
All these considerations could better be done in
the complex plane z x iy
If a
n
for n m a
m
one says that the point x
is a pole
a regular singular point of order m Singular points that are not poles are
called irregular singular or essential singular Equation is thus called
a Fuchs equation if xp
x and x
p
x are regular for x that means
that p
x has a pole of rst and p
x of second order respectively These
regular rational functions can be expanded
p
x
L
X
l
A
l
x a
l
p
x
L
X
l
B
l
x a
l
C
l
x a
l
L
X
l
C
l
For L we may write
p
x
x
X
n
n
x
n
p
x
x
X
n
n
x
n
According to Frobenius the singularity can be split o and the solution of
can be rewritten as a socalled Frobenius series
yx x
X
n
a
n
x
n
a
is called the index of the series The series is convergent Since the method
of power series is just the special case of the Frobenius method we
will discuss only the latter
We will now solve using the Frobenius method yields
y
x x
X
a
n
x
n
x
X
a
n
nx
n
y
x x
n
a
n
x
n
x
n
a
n
nx
n
x
X
n
nn an a
n
x
n
since a convergent power series may be dierentiated
Inserting and into and using we obtain
X
n
x
n
a
n
na
n
nn a
n
a
n
na
n
a
n
P
n
x
n
n
a
n
n
a
n
n
a
n
A power series vanishes only if all coecients vanish For n
reads
since a
cancels The case n will be treated later Equation
is the socalled indicial equation We now apply the method on a special form
of the Euler equation We use x
a b c so that
ba ca
x
y
x xy
x yx
Then the indicial equation reads
Its solutions are
so that the solution of is
given by the superposition of two particular solutions
yx AxBx
The command
DSolve[xˆ2*y[x]+3*x*y[x]-3*y[x]==0, y[x],x] delivers the same result It would be easy to show that this solution satises
for instance the initial conditions Also the boundary conditions
and can be satised by
In the case that the indicial equation has real repeated roots
or
n n the Frobenius method delivers only the rst
solution y
x The second solution will then contain an essential singularity
like a logarithmic term This solution may be derived from For
p
this equation reads
y
x y
x
Z
exp
R
p
xdx
dx
exp
Z
p
xdx
exp
lnx
x
x
x
P
x
where P
is a regular power series that does not vanish for x Assume
that the rst solution y
x has the form
y
x x
P
x
where P
x is a regular power series that does not vanish for x Then
the integrand in may be written in the form
y
x
exp
Z
p
xdx
x
P
x
P
xx
n
P
x
since
n Expanding the regular power series P
x
P
m
m
x
m
the integral becomes
Z
x
n
P
xdx
X
m
m
x
mn
m n
n
lnx m n
For x
a b
the Euler equation has the
solution y C
x
C
x
lnx For
the command
DSolve[xˆ2*y[x]+2*x*y[x]+2*y[x]==0, y[x],x] yields an expression containing power of x with complex exponents which is
equivalent to
yx x
C
cos
p
lnx
C
sin
p
lnx
If the roots of the indicial equation are complex they must be conjugate Then
the solution of may be expressed in terms of trigonometric functions
Up to now we have investigated only the case n We use the Bessel
equation
y
x
y
n
x
y
to demonstrate the procedure for n Inserting into we
receive for n replaced by
X
x
c
n
X
x
c
We can expect the existence of a logarithmic solution This solution and
the determination of c
will be discussed later on Making the replacement
we can join the two sums into one to receive
X
x
c
n
c
For we thus obtain the twotermed recurrence relation
c
c
c
n
c
is still unknown If we choose then yields
c
Furthermore we nd c
c
c
so that only
appears and the series representing the Bessel functions contains only the
power
The command
produces the solution
where the space replaces the representing multiplication As an exercise
the reader is invited to solve the following equations using the Frobenius
method and Mathematica
Equation Solution Indicial Equ
y
y y
a
sinx
y
x
y
n
x
y y J
n
ix I
n
x
n
y
x
y y C
x
C
x
y
yx
essential singularity x a
y
x
y
a
x
y y C
x
a
C
x
a
a
y
x
x
y
x
y c
c
y
x
y
x
x
y c
c
y
x
y
xy
c
c
y
xy
ny c
nc
Due to the recurrence formulae one has c
c
for and therefore
convergence
Apparently the singularities appearing in dierential equations help to clas
p
z
m
z a
m
z a
m
n
z a
n
p
z
A
A
z A
l
z
l
z a
m
z a
m
z a
n
m
n
then the equation is called a B
ocher equation Nearly all bound
ary value problems one may come across in physics engineering and applied
mathematics are of this type n l m
m
We now discuss some special cases Four singularities are to be found in
to
y
z
z a
z a
z a
y
z
A
A
z A
z
A
z
z a
z a
z a
yz
Heine equation one pole of rst order two poles of second order and one
pole at innity
y
z
z a
z a
y
a
a
q p p z
z
z z a
z a
y
Lam
e wave equation or Lam
e equation for for and
y
z a
z a
z a
y
A
A
z A
z
z a
z a
z a
y
Wangerin equation Three singularities are contained in to
y
z a
z a
y
A
A
z A
z
z a
z a
y
two poles of rst order in the nite domain and one pole of fourth order in
innity and in
y
z a
z a
y
A
A
z A
z
A
z
A
z
y
equation y q cos xy has three singularities two are essential
The hypergeometric equation is the grandmother of many equations used in
physics and engineering It reads
y
c a b z
z z
y
ab
z z
y
This important equation has poles at and The values of the index
are
c at the location x
c a b at x
and
a
b at innity The Legendre wave equation
y
z
z
y
a
z
pp
z
q
z
y
and the Legendre equation are children of exhibiting three
singularities
Two singularities are found in
y
z a
y
A
z
z a
y
and
y
z a
y
A
A
z
A
z
z a
y
but even the simple equation
y
a
z
y
y
y
z
a
has two poles at and Furthermore some wellknown and important equa
tions have two singularities the con uent hypergeometric equation Kummer
equation
y
c z
z
y
ay
which is a daughter of It has one pole at z and an essential
singularity at ! c is its indicial equation The Bessel
equation and
y
z
y
pp
z
y
the Bessel wave equation
y
z
y
z
q
p
z
y
as well as the generalized Bessel equation
y
y
z
p
y
y z
Z
p
z
Z
p
is a cylinder function like J
p
have also two singularities Other children
of are the Whittaker equation
y
z
z
y
solved by Whittaker functions or the Gegenbauer equation
y
z
zy
nn
z
y
One singularity will be found in
y
m
z a
y
A
z a
m
y
which comprises the Euler equation The most simple linear dierential
equation of second order is given by
y
The equation y
y
z a has a pole at z a and the solutions y
y
z a Also the Weber equation
y
q
p
q
z
y
which is a grandchild of has one pole but y
ky has an
essential singularity at z Also and have one singularity
This is a consequence of the Liouville theorem which expresses the fact that
all functions yz of a complex variable z x iy must either have one or
more singularities or be a constant
Problems
Now solve the initial value problems using
y,{x,0,Pi}] InterpolatingFunction ff gg
In order to plot the result we use now
Plot[Evaluate[y[x]/.%],{x,0,Pi}] In order to plot the values of yx must be known Evaluate replaces the denition of a new function as vx in problem of section
The phrase y[x]/.% has the meaning replace yx by the result of the last calculation ie the solution of the initial value problem
Now use Mathematica to solve the inhomogeneous boundary value prob
lem numerically for x
x
bsol=NDSolve[{y[x]+y[x]==0,y[0]==1.,y[2.]==2.}, y[x],{x,0,Pi}] Here we have given a name to the calculation Plotting is now possible
Now solve the homogeneous boundary problem
x
x
y
yx
y
yx
Clear[y]; ts=NDSolve[{y[x]+y[x]==0,y[0]==0,y[Pi]==0}, y[x],{x,0,Pi}] and plot the result If this does not work look at the values yx by
Find the indicial equation or for the following equations
solve them according to the Frobenius method and verify the result
with Mathematica Take some of the equations on page
DSolve[y[x]+mˆ2*y[x]==0,y[x],x] ffy x C Cos mx C Sin mxgg
DSolve[y[x]+y[x]/x-(1+nˆ2/xˆ2)*y[x]==0,y[x],x] ffy x BesselJ nix C BesselY nix C gg
modied Bessel function
y x x
C
C
x
y x
BesselI
p
q
x
C
p
q
x
q
BesselK
p
q
x
C
q
x
This does not work Why For numerical solution a range must be
given see problem
Solve the initial value problem of an equation of third order
Solve and giving
y x C HypergeometricF a b c x
c
x
c
C HypergeometricF a c b c c xgg
y x
exp a x C
a
C
y x C HermiteH
a
c
p
x
p
C HypergeometricF
a
c
p
x
p
y x C Cos x C Sin x
x
x Cos x
x Cos x CosIntegral x
x
Cos x CosIntegral x x CosIntegral x Sin x
x Sin x
x Cos x SinIntegral x
x Sin x SinIntegral x x
Sin x SinIntegral x
compare to
y[x],x] y x exp q
x
C
HermiteH
h
q
p q
q
q x
i
exp q
x
C
HypergeometricF
h
q
p q
q
q
x
i
Dierential equations of physics and engineering
Prior to the discussion of boundary problems it seems to be useful to inves
tigate some of the dierential equations of physics and engineering in more
detail A large class of partial dierential equations allows separation into
ordinary dierential equations Many of these ordinary dierential equations
are children or grandchildren of the hypergeometric dierential equation
In spherical problems the separation of the pertinent partial dierential
equation like eg Helmholtz equation leads to the Legendre
equation
d
y
d
cot
dy
d
ll y
m
sin
y
where is the polar angle in spherical coordinates The solutions of
are usually called spherical functions The substitution cos x gives rise to
the equation
y
x
x
x
y
x
ll
x
yx
m
x
yx
This is the special case of the Gegenbauer equation It is
easy to see that this equation has poles at the location x For m it
has the recurrence relation
c
c
l
l
c
is dened by The case m will be treated later With the
substitution x x one obtains from the new
equation
y
y
ll y
This equation has a pole at and is the special case a l b l c
transcendental spherical functions P
n
In order to write down the general
solution of we need a second solution But due to the relations
c and
cab which are valid for the hypergeometric
equation one obtains
for both poles This means that the
second solution is identical with P
n
Due to the mother the hypergoemetric
dierential equation its solutions are closely related So the function cos is the
elliptic and cosh the hyperbolic child Whereas the function of the rst
kind P
n
corresponds to cos the hyperbolic part is given by the Legendre
function Q
n
of the second kind
Q
n
x
Z
x
p
x
cosh t
n
dt x
This expression is a consequence of the possibility to represent the members
of the hypergeometric family by integrals see later The general solution of
the Legendre equation is now given by
yx C
P
n
x C
Q
n
x
If the parameter l is a natural number a positive integer then the P
n
de
generate into polynomials and the Q
n
go over into elementary transcendental
functions For n l the recurrence relation breaks down and the
solutions are given by the Legendre polynomials
P
x P
x x cos
P
x
x
cos
P
x
x
x
cos cos
These polynomials as well as other polynomials are important for physical
and engineering problems and may be represented by
a a Rodriguez formula
b using a generating function
c or by an integral representation
These possibilities oer many practical applications The Rodriguez for
mula for the Legendre polynomial is given by
P
l
x
l
l"
d
l
dx
l
x
l
and their generating function f is
fx u
ux u
X
P
l
xu
l
Cauchy integral
gz
i
I
C
gt
t z
dt
For gt t
n
one obtains
P
n
z
n
n
n"
d
n
dz
n
h
z
n
i
n
n
i
I
t
n
t z
n
dt
Using z x cos t cossin expi tx
i sin expi dt sin expi etc one obtains the Laplace integral
representation
P
n
cos
Z
cos i sin cos
n
d
Using the integral representation of the solution of the hypergeometric equa
tion the transcendental functions may be represented by
P
n
x
Z
x
p
x
cos t
n
dt x
compare
Up to now we have considered only the special case m For m
we consider The Frobenius method creates the solutions that are
called associated Legendre polynomials P
m
n
x and associated Legendre
functions respectively Mathematica denes all these various functions
P
l
x LegendreP[l,x] P
m
l
x LegendreP[l,m,x] P
n
z LegendreP[n,z] P
m
n
z LegendreP[n,m,z] Q
n
z LegendreQ[n,z] Q
m
n
but has not been able to produce these functions by solving equations
We give some formulae for the associated polynomials
P
m
x
m
d
lm
x
l
P
x
x
sin
P
x
x
x
sin
P
x
x
cos
P
x
x
x
sin sin
There exists an important attribute of these polynomials This feature is
very important for applications The attribute is called orthogonality and is
described by
Z
P
l
xP
k
xdx
l
lk
Z
P
m
l
xP
m
k
xdx for l k
Z
P
m
l
xP
m
l
xdx
l
l m"
l m"
Other children of the hypergeometric equation are the Chebyshev polyno
mials and Chebyshev functions if x is replaced by complex z
ChebyshevT[n,x] T n
x
ChebyshevU[n,x] U n
x
which satisfy the equation
x
T
n
x xT
n
x n
T
n
x
and are given by T
n
T
n
n
which xes the c
in their
recurrence relation and explicitely by
T
x T
x x T
x x
T
x x
x T
x x
x
T
x x
x
x
Chebyshev polynomials of rst kind are often used in making numerical
approximation to functions They satisfy the orthogonality relations
Z
T
m
xT
n
x
p
x
dx
for m n
for m n
for m n
The polynomials U
n
x of the second kind and the Chebyshev functions are
by Laguerre orHermite In contrast to the orthogonality relations
of the Legendre polynomials the orthogonality relations contain a
weighting function x
The same is true for the associatedLaguerre
polynomials They satisfy the Laguerre dierential equation
xL
k
n
x k xL
k
n
x nL
k
n
x
and the Rodriguez formula
L
k
n
x! L
n
x! k
e
x
x
k
n"
d
n
dx
n
e
x
x
nk
For k and the polynomials L
k
n
x are designated by L
n
x For
they are given by
L
x L
k
x
L
x x L
k
x x k
L
x
x
x L
k
x
x
k x
k k
L
k
n
x
X
m
m
n k"
nm"k m"m"
x
m
k n
The orthogonality relations again contain a weighting function They read
Z
e
x
x
k
L
k
n
xL
k
m
xdx
n k"
n"
mn
This has the important practical consequence that all functions fx that are
quadratic integrable can be expanded into a Laguerre series
fx
X
n
c
n
expxx
k
L
k
n
x
with
c
n
n"n k"
Z
fx expxx
k
L
k
n
xdx
Application of DSolve on yields the result in the form of a con u ent hypergeometric Kummer function For k one obtains an
analogous result but for n the results are retrieved
Additionally a transcendental function appears that seems to have no prac
tical applications
Hermite polynomials H
n
x have similar important properties They sat
isfy the Hermite equation
hypergeometric function Solution by the Frobenius methods leads to the
recurrence relation
c
c
n
and the Rodriguez formula reads
H
n
x!