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May 24th

May 24th

# The General, Linear Equation

PROBLEM SET 2.1.2

1. For the equation u '' + u = x, find a particular integral by inspection. What
is the most general solution? What is the solution with initial data (u, u') =
(0, 0) at x = 0?

2. Same question as above but for the equation u'' − u = x.

3. The inhomogeneous form of the self-adjoint equation (cf. Problem Set 2.1.1,
exercise 15) is

In it, the coefficient of the highest derivative is not one, but p. Ex press a
particular
integral of this equation in terms of a solution basis , for the
homogeneous equation and the function r, as in equation (2.24) above.

4. Carry out the construction of the influence function for the equation u''+u =
r.

5. Verify that the expression for the influence function given in equation (2.25)
satisfies the initial-value problem (2.27).

6. Verify directly that if G satisfies the initial-value problem (2.27), then
the expression (2.26) provides a particular integral.

7. Consider the homogeneous equation u'' + q(x)u = 0 where

Require of solutions that they be C1 for all real values of x (i.e., continuous
with continuous derivatives: the second derivative will in general fail to exist
at x = 0). Construct a basis of solutions. Check its Wronskian.

8. For q as in Problem 7, find the solution of the initial-value problem

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

9. For the differential equation of Problem 8, find the influence function G(x, s).

2.2 The equation of order n

The results of the preceding section carry over to higher-order linear equations
with remarkably few modifications. The linear operator L is now given
by equation (2.1), and the homogeneous equation by equation (2.3). As in
the case of the second-order equation, we consider first the existence theorem
for the initial-value problem

where is any point of the interval [a, b] and the numbers are
arbitrarily chosen constants.

Theorem 2.2.1
Suppose the functions and r are continuous
on [a, b]. Then the initial-value problem (2.31) has a unique solution u (x) on
that interval.

Remarks

• The existence result is again borrowed from Chapter 6.

• The remarks fol lowing theorem (2.1.1) apply here as well.

• The uniqueness result is incorporated in this theorem instead of being
stated separately. Its proof may be carried out in the same way as in
the second-order case except that the definition of in the proof of
Lemma (2.1.1) is changed to

and the manipulations are somewhat more cumbersome.

• Theorem 2.1.4 carries over without change (see Problem 10 in Problem
Set 2.3.1 below).

It is again convenient to discuss the homogeneous equation first.

2.2.1 The homogeneous equation

Since in place of two arbitrary constants in the solution of a second-order
equation we now anticipate n arbitrary constants in the solution of an n-th
order equation, we must also anticipate that n linearly independent solutions
are needed to represent an arbitrary solution of the equation (2.3). Accordingly,
we define the Wronskian of n functions on an interval [a, b] as follows:

This definition is valid for any n functions defined and sufficiently differentiable
on [a, b]. When these functions are solutions of equation (2.3), the
Wronskian serves the same purpose that it did in the second-order case, by
virtue of the following theorem:

Theorem 2.2.2

Remarks:

• This theorem takes exactly the same, simple form for a linear, homogeneous
differential equation of any order: the coefficient of the term
in equation (2.1) is what appears in the exponential .

• Corollary 2.1.1 applies almost without change to the Wronskian (2.32)
for the n-th order case:

Corollary 2.2.1 The Wronskian of a set of n solutions of a linear,
homogeneous equation (2.3) of order n either vanishes identically on
the entire interval [a, b] or does not vanish at any point of the interval.

• The proof of Theorem 2.2.2 depends on a formula for the differentiation
of determinants; it is discussed in the Problem Set 2.3.1 below.

If we choose a matrix A with ij entry , i.e.,

we can construct n solutions of equation (2.3) whose Wronskian,
at a chosen point of [a, b], is precisely detA. It suffices to choose
to be the solution of equation (2.3) such that

These solutions exist by the existence theorem 2.2.1. They are linearly independent
on the interval [a, b] if we choose A so that detA ≠ 0. This can be
d one in many ways.

Example 2.2.1 Choose A to be the unit matrix, i.e. = 0 if i ≠ j and
= 1 for i, j = 1, . . . n. Then it is easy to see that detA = 1.

Theorem 2.1.7 is fundamental. It asserts that once a set of solutions
of equation (2.3) that have non zero Wronskian have been found, any other
solution of that equation is a linear combination of them . We now have the
same result in general:

Theorem 2.2.3 A necessary and sufficient condition for a set
of solutions of equation (2.3) to be a basis for that equation is that its Wronskian
W not vanish on [a, b].

Example 2.2.2 Consider the third-order equation u''' = 0. Three linearly
independent solutions are = 1, = x and . It is easy to see that
any solution of the equation must have the form .

Example 2.2.3 Consider the third-order equation u''' + u' = 0. If v = u'
the equation becomes v'' + v = 0, which has linearly independent solutions
= cos x and = sin x. Choose such that . Then =
sin x + constant. Similarly choosing such that , we find that
= −cos x + constant. Since sin x, cos x and any constant are solutions of
this equation, we easily see that {1, cos x, sin x} constitute a basis of solutions.

2.2.2 The inhomogeneous equation

The two methods used to produce a particular solution of the inhomogeneous
equation can be carried out with minor changes. The variation-of-parameters
expression is now

where represent a basis of solutions of the homogeneous equation. It
can be supplemented by n − 1 conditions on the as-yet-unknown functions
. These are chosen in the form

where the convention = u has been used. The condition that u satisfy
equation (2.4) is then that

These give n equations for the n functions . The determinant of this system
of equations is the Wronskian, so it can be solved for the derivatives of the
. If they are all integrated from (say) to an arbitrary point x of the
interval with the as sumption that () = 0, then the particular integral u
found in this way will vanish together with its first n − 1 derivatives at .
The formula is now more cumbersome to write, so we omit it.

The alternative approach to a particular integral via an influence function
is the following. Define the influence function G(x, s) for a ≤ s ≤ x ≤ b
by the following initial-value problem, where s is viewed as fixed and the
derivatives are taken with respect to x:

On the remainder of the square [a, b]×[a, b] where s > x, G vanishes. It can
be verified directly that this provides a particular integral of equation (2.4)
in the form

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