Linear differential equations
Additivity of solutions of linear diff. equations;
Homegeneous and non-homogeneous linear ODE's,
their general and particular solutions.
d
dt
Eigenfunctions of the differentiation operator ,
general solution of a homogeneous linear ODE using
the method of auxiliary equation.
Revision of properties and operations with of complex
numbers,
drawing complex numbers on a 'complex plane',
modulus and the phase of a complex number.
Functions of complex variables and the function
e
it
ODE’s describing damped linear oscillators.
Literature:
All you wanted to know about Mathematics,
but were afraid to ask, L.Lyons - Vol. 1, Chapts. 4,5
Other reading:
Mathematical Methods for Science Students, G.Stephenson – Chapts.7,20
Mathematical Methods in Physical Sciences, M.L.Boas – Chapts. 2,8
Linear differential equations with constant
coefficients (definitions)
1st order
dx(t )
 k0 x(t )  f (t )
dt
2nd order
d 2x
dx
 k1
 k0 x  f (t )
2
dt
dt
nth order
d nx
d n1 x
dx
 kn1 n1      k1  k0 x  f (t )
n
dt
dt
dt
all
k0 , k1 ,  , kn1
are constants (independent of the variable
t)
__________________________________
homogeneous
d 2x
dx
 k1  k0 x  0
2
dt
dt
non-homogeneous
d 2x
dx
 k1  k0 x  f (t )
2
dt
dt
__________________________________
Differential equation of the nth order may be
complemented with n initial (or boundary) conditions:
0t 
dx(0)
x(0)  x0 ;
 v0
dt
Initial conditions
Boundary conditions
0t T
x(0)  A; x(T )  B
Linear ODE’s represent an important class of differential equations since:
*)
they are encountered in various physics problems
Kinetic equation for
radioactive decay
dx 1
 x0
dt 
Linear oscillator

d 2x k
 x0
2
 x0
dt
m
**)
more sophisticated problem problems can be approximated by linear
equations (linearisation)
***)
they have several useful properties which allow one to find a
general solution of any linear ordinary differential equation.
_______________________________________________________________________
Property of additivity of solutions:
is a solution of
d 2x
dx
 k1  k0 x  f1 (t )
2
dt
dt
and
Y (t ) is a solution of
d2y
dy
 k1  k0 y  f 2 (t )
2
dt
dt
then
Z (t )  X (t )  Y (t )
is a solution of a linear diff. equation
if
X (t )
d 2z
dz
 k1  k0 z  f1 (t )  f 2 (t )
2
dt
dt
_______________________________________________________________
If x1 (t ) and
then
x2 (t ) are two solutions of a homogeneous linear ODE
d 2x
dx

k
 k0 x  0
1
2
dt
dt
x(t )  x1 (t )  x2 (t ) is also a solution of the same ODE.
Example: If X A A (t ) is the general solution of a homogeneous linear ODE
1 2
d 2x
dx

k
 k0 x  0
1
2
dt
dt
and Y (t ) is some particular solution of a non-homogeneous linear
ODE with the same coefficient,
d2y
dy

k
 k 0 y  f (t )
1
2
dt
dt
then X A1 A2 (t )  Y (t ) is the general solution of a
non-homogeneous equation
d 2x
dx

k
 k 0 x  f (t )
1
2
dt
dt
This enables us to split a harder problem of finding a general
solution of a non-homogeneous equation into two simpler ones:
1) to find a general solution of a homogeneous equation
d 2x
dx

k
 k0 x  0
1
2
dt
dt
2) to find any particular solution of an original
non-homogeneous equation.
_______________________________________________________________________
The general solution of any homogeneous linear differential
equation can be found using the fact that exponential functions
are the eigenfunctions of the differentiation operator
differentiation
operator

d Dt
Dt
e  De
dt 

Eigenfunction
(proper function)
Eigenvalue
For exponential functions,
d Dt
Dt
e

D

e
the action of the differentiation
dt
operator is equivalent to the d
Ae Dt  D  Ae Dt
multiplication by a number:
dt






d2
d
Dt
Dt
2
Dt
Ae

D

Ae

D

Ae
dt 2
dt
n 1
dn
d
Dt
Dt
n
Dt
Ae

D

Ae





D

Ae
dt n
dt n1




_________________________________________________________
For
x(t )  Ae Dt
with some
D
d nx
d n1 x
dx
 k n1 n1      k1
 k0 x  0
n
dt
dt
dt
D
n
 kn1D
n1


     k1D  k0  AeDt  0
Dt
e
0
since

auxiliary
equation
D n  k n1 D n1      k1 D  k0  0
When this algebraic equation has n different roots D1 , D2 ,  , Dn
the general solution of the homogeneous linear ODE has the form
x(t )  A1e
where
D1t
 A2 e
D2t
     An e
A1 , A2 ,  , An are free parameters (one can find them using
Dn t
initial / boundary conditions).
Example:
2nd order linear ODE with constant coefficients
d 2 x dx
  2x  0
2
dt
dt

substitute
x(t )  AeDt
[ D 2  D  2]  Ae Dt  0
auxiliary
1
1
2
D

D


2

D  D  2  0 equation
4
4
1
9
3 1
3 1
( D  )2 
 D1    2, D2     1
2
4
2 2
2 2
2
x(t )  A e  A e
2t
t
General solution
1
2
_______________________________________________
Differential equation + Boundary conditions
d 2 x dx
  2x  0
2
dt
dt
0t T
x(0)  0; x(T )  1
e0  1
20
0

x
(
0
)

A
e

A
e
 A1  A2  0

1
2


2T
T

A2   A1
 x(T )  A1e  A2e  1
A1e
2T
 A1e
T
Particular solution
for given boundary conditions:
1 
A1 
e 2T
e 2t  e  t
x(t )  2T
e  e T
1
 e T
Example: Ball rolling of the top of a hill
d 2 x Force k

 x
2
dt
m
m
x(0)  0;
dx(0)
 v0
dt
2
d x k
 x0
2
dt
m
F ( x)  
v
k
[ D  ]  Ae Dt  0
m
2
D1 
k
k
, D2  
m
m
x(t )  A1e
0
kx2
U ( x)  
2
General solution
t k/m
dU ( x)
 kx
dx
 A2e
t k / m



h1  A1  A2
h2  A1  A2

 h1 cosh t k / m  h2 sinh t k / m
e Dt  e  Dt
cosh Dt 
2
cosh( 0)  1
e Dt  e  Dt
sinh Dt 
2
sinh( 0)  0

x(0)  h1 1  h2  0  h1  0
dx(0)
 Dh1  0  Dh2  1  v0
dt
Particular solution

d
cosh Dt  D sinh Dt
dt
d
sinh Dt  D cosh Dt
dt

h1  0
v0
h2 
D

v0
x(t ) 
sinh t k / m
k /m

kx2
U ( x) 
2
Example: Linear oscillator

2
d x Force
k

 x
2
dt
m
m
x
d 2x k
 x0
2
dt
m
F  kx Hooke’s

law
k
[ D  ]  Ae Dt  0
m
2
k
D 
m
2
auxiliary
equation
k
k
D1  i
, D2  i
m
m
to solve this, one has
to use complex numbers
Since, in general, an auxiliary equation
D 2  k1 D  k0  0
corresponding to some 2nd order linear differential equation
d 2x
dx
 k1
 k0 x  0
2
dt
dt
may have complex roots, to solve it, one has to know how
to operate with complex numbers and also with function
e it .