# The General, Linear Equation

**2.1.2 The Wronskian
**

The Wronskian determinant of a pair of functions is defined as follows:

**Definition 2.1.3**

We often write W (u, v; x) = W (x) for short. When we do
this we need to

remember that we have a specific pair of functions u and v in mind, and that

the sign of the Wronskian changes if we interchange them.

**Theorem 2.1.6** Let u and v be solutions of equation (2.7). If
is any

point of [a, b] and the value of the
Wronskian there, then, at any point x

of [a, b],

Proof: Differentiating W as defined in equation (2.16)
gives W' = uv''−u''v.

Substituting for the second derivatives from the equation (2.7), satisfied by

each of u and v, then gives W' = −pW. The conclusion follows from this.

An immediate deduction from this is the following:

**
Corollary 2.1.1 **The Wronskian of a pair of solutions of a linear,
homogeneous

equation (2.3) either vanishes identically on the entire interval [a, b]

or does not vanish at any point of the interval.

The Wronskian is precisely the determinant appearing in the systems

(2.13) and (2.14) above. When the Wronskian vanishes, the solutions and

are linearly dependent, and when the Wronskian does not vanish they

are linearly independent and therefore constitute a basis. This proves the

following:

**Theorem 2.1.7**A necessary and sufficient condition for a pair of

solutions of equation (2.7) to be a basis for that equation is that its Wronskian

W not vanish on [a, b].

Remark: If a pair of solutions
and
is chosen “at
random,” i.e., if

their initial data are assigned randomly, we would expect that W ≠ 0; in

this sense it would take a “peculiar” choice of solutions for that pair to be

linearly dependent.

The expression (2.17) for the Wronskian may be used to find a formula

for a second solution v of equation (2.7) if we know one solution u. Suppose

one such solution u (not identically zero) is known on [a, b]. Pick a point

where u ≠ 0 and define

where c is not zero but is otherwise arbitrary. Write

so that

This formula defines v on an interval containing on
which u does not vanish.

It is straightforward to check that v is indeed a solution of equation (2.7)

on such an interval, and that the pair u, v has the nonvanishing Wronskian

w there.

PROBLEM SET 2.1.1

1. Verify equation (2.2) in detail for the special case when n = 2 and the

operator L is defined by equation (2.5). Using equation (2.2), verify Lemma

2.0.2.

2. Let L be the second-order operator defined by equation (2.5). Suppose k

functions (x) , i = 1, . . . , k are given on the interval [a, b] and the
corresponding

k initial-value problems

are solved for the k functions
, i = 1, . . . , k. Define
.

Show that the unique solution of the initial value problem Lu = r, u () =

u' () = 0 is

3. In equation (2.7) let the interval be [-1, 1]. Can
continuous coefficients q

and p be chosen such that u (x) = x^{2} is a solution of this equation?

4. Let u (x) = x^{3} on [-1, 1] and define v (x) = -x^{3} on
[-1, 0] and v (x) = +x^{3}

on (0, 1]. Verify that v is C^{2} on [-1, 1]. Calculate W (u, v; x). Are these

functions linearly dependent on [-1, 1]?

5. The fundamental theorem of algebra states that a polynomial of degree n

can have no more than n distinct zeros. Draw the conclusion of Example

2.1.1 from this.

6. Show that the three functions sin(x), sin(2x), sin(3x) are linearly
independent

on any nontrivial interval of the x axis.

7. For the equation u''+u = 0, cos x and sin x form a basis of solutions. Verify

that cos (x + a) and sin (x + a) are also solutions if a is any constant. Use

this and the uniqueness theorem to infer the trigonometric addition theorems

cos (x + a) = cos a cos x - sin a sin x,

sin (x + a) = cos a sin x + sin a cos x.

8. Find bases of solutions for the following equations:

a) u'' = 0;

b) u'' + 2u' = 0;

c) u'' + xu' = 0.

9. Consider the equation u'' + u = 0. Under what conditions on the constants

a, b, c, d is the pair

a basis of solutions for this equation?

10. Consider the equation u'' - u = 0. Under what conditions on the constants

a, b, c, d is the pair

a basis of solutions for this equation?

11. Show that the functions and
form a basis of

solutions for the equation

on any interval excluding the origin.

12. Same as the preceding problem for the functions
=
sin x − x cos x ,
=

cos x + x sin x and the differential equation

13. Verify that the second solution given in the formula
(2.18) is linearly independent

of u on intervals on which u does not vanish.

14. Let u be a solution of (2.7) vanishing at the endpoints
and of an

interval but not vanishing in between. Show that the formula (2.18), with

, leads to finite limiting values for the
second solution v at

and , and find these limits.

15. A second-order equation

is said to be in ”self-adjoint” form. What condition on
the function p is

needed for the existence theorem 2.1.1 to hold? Show that the Wronskian

of such a self-adjoint equation is, except for a multiplicative constant, p^{-1}.

16. For the equation

• On what interval does the existence theorem guarantee a
solution?

• Verify that = x is a solution.

• Find a second solution in an interval containing the origin. How far

can this interval extend on each side of the origin?

17. Let u and v be C^{2} functions on an interval I, and suppose their Wronskian

uv' − u'v does not vanish on I. Show that their zeros separate each other,

i.e., between any two consecutive zeros of u there is exactly one zero of v

and vice versa.

18. Consider the equation u'' +p(x)u' +q(x)u = 0 where p and q are continuous

on the entire real axis R. Suppose further that q(x) < 0 there. Show that if

u is not identically zero, it can have at most one zero on R.

19. Let u and v be given C^{2} functions on an interval [a, b] whose Wronskian

nowhere vanishes there. Show that there is a differential equation of the

form (2.7) for which u and v form a basis of solutions.

20. Consider the initial-value problem on the interval [−1, 1]

u'' + q (x) u = 0, u (−1) = 1, u' (−1) = 0,

where the coefficient q is the sectionally continuous function

Find a function u satisfying the initial conditions, C^{1} on
the interval [−1, 1],

and satisfying the equation on the intervals [−1, 0) and (0, 1].

21. Suppose p and q are continuous functions on the symmetric interval [−a, a]

and satisfy the further conditions

p(−x) = −p(x) and q(−x) = q(x)

there; i.e., p is an odd function and q is even. For the differential equation

u'' + p(x)u' + q(x)u = 0 show

(a) If u(x) is a solution then so is v(x) = u(−x).

(b) There exists a basis of solutions , of which one is even on the

interval, the other odd.