Aspectos éticos

It has been mentioned from time to time that solutions of differential equations are often subject to extra conditions. This section will be devoted to a discussion of two types of conditions that are of frequent occurrence in practice.

Suppose a solution of

(3.6.1) is required such that

The general solution of (3.6.1) is

where yp is a particular integral and the remaining terms represent the complementary function. Then the imposed conditions can be complied with if

when t=t0. These n equations for C1,…, Cn can always be solved if

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This determinant is the same as that derived from the Wronskian of Section 3.3 when the differential equation is converted to a first-order system and, accordingly, is also known as a Wronskian. The non-vanishing of the Wronskian warrants the statement that

(3.6.2)

for an interval of t, which necessitates the constants B1,…, Bn all being zero. For n−1 derivatives of (3.6.2) give a set of equations for B1,…, Bn with non-zero determinant and zero right-hand side. Expressed in other words, the non-vanishing of the Wronskian makes y1,…, ynlinearly independent over the interval. To put it another way, it makes certain that the complementary function has been determined correctly.

Since there is only one set of C1,…, Cn that satisfies the equations for non-zero Wronskian, it has been demonstrated that the initial value problem always possesses a solution and there is only one which satisfies the imposed conditions. This constitutes another verification of the uniqueness property.

The initial value problem is characterised by all the restrictions being applied at a single value of t. In some instances the conditions refer to more than one value of t—we then have a boundary value problem. In contrast to the initial value problem, it is by no means certain that a boundary value problem has a solution. Consider

of which the general solution is

Let the conditions be y(0)=0, y(1)=0. The first requires C1=0 and the second C2 sin 1=0. Since sin 1≠0 we must have C2=0 and the only solution is the trivial one which vanishes everywhere. Now change the

conditions to y(0)=0, y(π)=0. In this event, y=C2 sin t is a solution with C2 arbitrary. Thus, boundary value problems may have many solutions or none (if the trivial one is discounted). It is also obvious that the interval of t has a critical role to play.

Instead of varying the interval, it is usual to fix it and incorporate a parameter in the differential equation. A typical problem might be to solve

(3.6.3)

subject to y(a)=0, y(b)=0. The values of λ, which is independent of t, are crucial. For some there will be only the trivial solution and for others there will be many solutions. Those λ for which non-trivial solutions exist are called eigenvalues and the corresponding solutions eigenfunctions.

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Page 68 Example 3.6.1 Consider

under the conditions y(0)=0, y(π)=0.

If λ=0, the general solution is y=A+Bt, which satisfies the boundary conditions only if A=0 and B=0. Therefore λ=0 is not an eigenvalue.

To comply with the boundary conditions we must have C1=0 and For a non-trivial solution C2≠0 and λ=m2 where m is a positive integer. The eigenvalues are real and infinite in number. They may be designated λ1, λ2,… where λm=m2. The eigenfunction corresponding to λm is Cm sin mt where Cm is

arbitrary.

The discussion of (3.6.3) will assume that p(t) is a continuously differentiable real function that does not change sign for any t in (a, b). No loss of generality is incurred in taking it to be positive. The function q will be assumed to be real and continuous in (a, b). We shall also suppose that there is an infinite set of

eigenvalues λ1, λ2,…with associated eigenfunctions Y1, Y2,….

With these assumptions the first thing to be shown is that the eigenvalues are real. Ym satisfies (3.6.4) and Ym(a)=0, Ym(b)=0. By taking a complex conjugate

(3.6.5)

and Multiply (3.6.4) by , (3.6.5) by Ym and subtract. There results

Hence

(3.6.6)

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by integration by parts. The right-hand side of (3.6.6) is zero because of the conditions on Y, Y* at t=a, t=b. The integral on the left is positive because Ym is a non-trivial solution. Consequently, and λm is real. The reality of the eigenvalues means that there is no loss of generality in taking the eigenfunctions to be real. With this understood note that

(3.6.7) and Yn(a)=0, Yn(b)=0. Multiply (3.6.4) by yn, (3.6.7) by Ym, subtract and proceed as above. Then

(3.6.8) The right-hand side is zero on account of the values of Ym, Yn at the endpoints. Therefore, if λm≠λn,

(3.6.9)

Functions that satisfy (3.6.9) are said to be orthogonal, i.e., the eigenfunctions of distinct eigenvalues are orthogonal. If, in addition, the eigenfunctions are normalised so that the eigenfunctions are called orthonormal.

It will not have escaped the reader’s notice that the right-hand sides of (3.6.6) and (3.6.8) can vanish for conditions other than those delineated. For example, if y(a)=0 is replaced by

(3.6.10)

where at least one of the real α1, α2 is non-zero, the right-hand sides are still zero. A similar remark is true if y(b)=0 is changed to

(3.6.4)

where β1, β2 are real with at least one non-zero. Equations (3.6.10) and (3.6.11) can be deemed standard

boundary conditions (they include the previous ones by putting α2=0, β2=0). What has been shown is that the eigenvalues are real and the eigenfunctions orthogonal for standard boundary conditions.

Example 3.6.2

Find the eigenfunctions of ÿ+λy=0 subject to , .

For λ≠0, proceed as in Example 3.6.1 to show that there is an eigenvalue n2 with eigenfunction Cn cos nt.

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The first eigenfunction can be subsumed in the second group by allowing n=0. Thus the eigenfunctions are Cn cos nt for n=0, 1, 2,….

Eigenvalues can also occur for periodic boundary conditions where p(a)=p(b), y(a)=y(b), . Again the eigenvalues are real and the eigenfunctions orthogonal.