ORDINARY DIFFERENTIAL EQUATION
SOME BASIC DEFINITIONS
Definition: A function is said to be analytic in a neighbourhood I = {x : |x − x0 | < r0 }
of a point x0 , if derivatives of all order exist, for the function f (x), in I i.e., if f (x) also
possesses a convergent infinite Taylor series expansion or power series expansion in I.
The function f (x) has a power series expansion implies f (x) = ∞ i
P
i=0 ai (x − x0 ) , where ai s
′
are constants. The series on the right hand side converges to f (x) in I.
f ′ (x0 ) f n (x0 )
It may be noted that a0 = f (x0 ), a1 = 1!
, ..... an = n!
.
So we have
′ (x
0)
n (x
0)
f (x) = f (x0 ) + (x − x0 ) f 1!
+ .... + (x − x0 )n f n!
+ .....
i.e., f (x) has infinite Taylor’s series expansion in I. In other words, f (x) possesses deriva-
tives of all order in I.
Linear Dependence, Linear Independence and Wronskian:
Definition: The functions fi (x), i = 1, 2, ...n are saidPto be linearly dependent if con-
stants c1 , c2 , ...cn , not all zero, can be found such that ni=0 ci fi (x) = 0.
Otherwise,
Pn the functions fi (x), i = 1, 2, ...n are said to be linearly independent if
c f
i=0 i i (x) = 0 implies ci = 0 ∀i = 1, 2, ...n.
Definition: Let the functions fi (x), i = 1, 2, ...n be n real functions, each of which has
(n − 1) derivatives in I then the determinant
f1 f2 ... fn
f1 ′ f2 ′ ... fn ′
. . . .
∆(f1 , f2 , ....fn ) ≡ W (f1 , f2 , ....fn ) = . . . .
. . . .
. . . .
f1 (n−1) f2 (n−1) ... fn (n−1)
is called wronskian of fi (x), i = 1, 2, ...n.
Theorem: A necessary and sufficient condition for linear dependence of two analytic
functions f1 (x), f2 (x) for x ∈ I is ∆(f1 , f2 ) = 0 in I.
Condition necessary:
If f1 (x), f2 (x) are linearly dependent then we can find constants c1 , c2 not simultaneously
zero such that
c1 f1 (x) + c2 f2 (x) = 0
1
which on differentiation yields
c1 f1 ′ (x) + c2 f2 ′ (x) = 0
Since c1 , c2 do not vanish simultaneously, so ∆(f1 , f2 ) = 0 in I.
Condition sufficient:
f1 f2
Let ∆(f1 , f2 ) = 0 in I. This implies = 0 =⇒ f1 f2 ′ − f2 f1 ′ = 0.
f1 ′ f2 ′
Now two cases may arise.
Case I: f1 ≡ 0. Then we have c1 f1 ′ (x) + 0.f2 ′ (x) = 0 in I. This implies f1 (x), f2 (x) are
linearly dependent.
Suppose f1 is not identically zero in I and let E denotes the set of zeros of f1 in I.
′ ′
Thus in I − E, f1 6= 0. Therefore in I − E, f1 f2f−f1
2
2 f1
= 0 =⇒ dxd
(f2 /f1 ) = 0 =⇒
f2 = Af1 , A being a constant.
Since f1 (x), f2 (x) are differentiable everywhere in I, so, f2 = Af1 in I. This shows that
f1 (x), f2 (x) are linearly dependent.
This theorem can be extended to the case of n functions fi (x), i = 1, 2, ...n.
Theorem: A necessary and sufficient condition for linear dependence of n analytic func-
tions f1 (x), f2 (x).....fn (x) for x ∈ I is ∆(f1 , f2 ...fn ) = 0 in I.
Note that the above theorems are true for analytic function. However in general we have
the following theorem.
Theorem: If wronskian of 2 functions f1 (x), f2 (x) defined on I is nonzero at atleast one
point of I, then the set of 2 functions are linearly independent in I.
Proof: Let f1 (x), f2 (x) be linearly dependent in I. Then there exists constants c1 , c2 not
all zero such that
c1 f1 (x) + c2 f2 (x) = 0 (1)
which on differentiation yields
c1 f1 ′ (x) + c2 f2 ′ (x) = 0. (2)
As per given condition, there exists a point x0 ∈ I such that
∆(f1 , f2 )(x0 ) 6= 0 (3)
f1 (x0 ) f2 (x0 )
in I. This implies 6= 0 =⇒ f1 (x0 )f2 ′ (x0 ) − f2 (x0 )f1 ′ (x0 ) 6= 0.
f1 ′ (x0 ) f2 ′ (x0 )
Thus from (1) and (2), we have
c1 f1 (x0 ) + c2 f2 (x0 ) = 0
2
c1 f1 ′ (x0 ) + c2 f2 ′ (x0 ) = 0.
f1 (x0 ) f2 (x0 )
Since c1 , c2 are not zero simultaneously, so =0
f1 ′ (x0 ) f2 ′ (x0 )
which is a contradiction by (3). Thus our assumption that f1 (x), f2 (x) are linearly depen-
dent in I is wrong. Hence the theorem.
Theorem: If wronskian of n functions f1 (x), f2 (x), ....fn (x) defined on I is nonzero at
atleast one point of I, then the set of n functions are linearly independent in I.
However, the converse is not true. the Two or more functions can be linearly independent
yet their wronskian may be zero.
Example: f1 (x) = x2 , f2 (x) = x|x|, − ∞ < x < ∞.
f2 (x) = x2 , for x > 0,
= 0, for x = 0
= x2 for x < 0.
f2 ′ (x) = 2x, for x > 0,
= 0, for x = 0
= −2x for x < 0.
fi (x), fi ′ (x), i = 1, 2 are continuous P
in (−∞, ∞). Also, fi (x), i = 1, 2 are linearly indepen-
dent. For this we have to show that ni=0 ci fi (x) = 0 =⇒ c1 = c2 = 0.
For x > 0, c1 f1 + c2 f2 = 0 =⇒ c1 x2 + c2 x2 = 0 =⇒ x2 (c1 + c2 ) = 0 =⇒ c1 = −c2 =
0, since x > 0. Thus, ∀x c1 x2 + c2 x|x| = 0 =⇒ c1 = 0, c2 = 0.
x2 x|x|
Now, ∆(f1 , f2 ) = = 0, ∀x.
2x 2|x|
This shows that ∆(f1 , f2 ) is identically equal to zero in (−∞, ∞) but f1 and f2 are linearly
independent.
It may be noted here that although f1 (x) is analytic in (−∞, ∞) but f2 (x) is not analytic
at x = 0. For
f2 ′′ (x) = 2, for x > 0,
= 0, for x = 0
= −2 for x < 0.
Now we come to the context of ordinary differential equations.
Theorem: Let y1 , y2 be two analytic solutions of ODE L1 (y) = 0 in I. Let x0 be any point
in I. Then
3
Rx
− a1 (t)dt
∆(x) = ∆(x0 ) e x0
Proof:
The analytic functions y1 , y2 satisfy the ODE L1 (yi ) = 0 in I.
The wronskian of y1 , y2 in I is given by
y1 y2
∆(y1 , y2 ) = = y1 y2 ′ − y2 y1 ′ .
y1 ′ y2 ′
Now,
y1 y2 y1 y2
∆′ (y1 , y2) = = y1 y2 ′′ − y2 y1 ′′ =
y1 y2
′′ ′′
−a1 y1 − a2 y1 −a1 y2 ′ − a2 y1
′
y1 y2 y y
= −a(x) − a2 (x) 1 2 = −a1 (x)∆(x).
y1 ′ y2 ′ y1 y2
Thus
∆′ (t) = −a1 (t)∆(t).R
This is a first order ODE for ∆(t). Solving we have
− xx a1 (t)dt
∆(x) = ∆(x0 ) e 0 .
Remark 3: Suppose ∆(x0 ) 6= 0 where x0 is any arbitrary point in I. Then (y1 , y2) are
linearly independent. Moreover since x0 is any arbitrary point in I so if ∆(x0 ) 6= 0 then
∆(x) never equal to 0 in I. Thus we can say that the analytic functions y1 , y2 which are
solutions of L1 (y) = 0 are linearly independent iff ∆(x) 6= 0, ∀ x ∈ I.
Definition: The set of linear independent solutions {y1 , y2 , ....yn } of nth order ODE
L(y) = 0 is known as Fundamental set.
Any solution y(x) of L(y) =P0 can be written as linear combination of the elements of
fundamental set, i.e., y(x) = ni=0 ci yi (x).
