Download (direct link):
y" + P(x )y'+ q (x )y = 0 (13)
and the boundary conditions
y(a) = 0, y(fi) = 0. (14)
Chapter 10. Partial Differential Equations and Fourier Series
Observe that this problem has the solution y = 0 for all x regardless of the coefficients p(x) and q (x). This solution is often called the trivial solution and is rarely of interest. What we usually want to know is whether the problem has other, nonzero, solutions. Consider the following two examples.
Solve the boundary value problem
y" + 2 y = 0, y (0) = 0, y (n) = 0. (15)
The general solution of the differential equation is again given by Eq. (8),
y = c1 cos V2x + c2 sin V2x.
The first boundary condition requires that c1 = 0 and the second boundary condition leads to c2 sin = 0. Since sin V2n = 0, it follows that c2 = 0 also. Consequently, y = 0 for all x is the only solution of the problem (15). This example illustrates that a homogeneous boundary value problem may have only the trivial solution y = 0.
Solve the boundary value problem
y" + y = 0, y (0) = 0, y (n) = 0. (16)
The general solution is given by Eq. (11),
y = c1 cos x + c2 sin x,
and the first boundary condition requires that c1 = 0. Since sin n = 0, the second boundary condition is also satisfied regardless of the value of c2. Thus the solution of the problem (16) is y = c2 sin x, where c2 remains arbitrary. This example illustrates that a homogeneous boundary value problem may have infinitely many solutions.
Examples 1 through 4 illustrate (but of course do not prove) that there is the same relationship between homogeneous and nonhomogeneous linear boundary value problems as there is between homogeneous and nonhomogeneous linear algebraic systems. A nonhomogeneous boundary value problem (Example 1) has a unique solution and the corresponding homogeneous problem (Example 3) has only the trivial solution. Further, a nonhomogeneous problem (Example 2) has either no solution or infinitely many and the corresponding homogeneous problem (Example 4) has nontrivial solutions.
Eigenvalue Problems. Recall the matrix equation
Ax = Xx (17)
that was discussed in Section 7.3. Equation (17) has the solution x = 0 for every value of X but for certain values of X, called eigenvalues, there are also other nonzero solutions, called eigenvectors. The situation is similar for boundary value problems. Consider the problem consisting of the differential equation
y" + Xy = o,
10.1 Two-Point Boundary Value Problems
together with the boundary conditions
y(0) = 0, y(n) = 0. (19)
Observe that the problem (18), (19) is the same as the problems in Examples 3 and 4 if X = 2 and X = 1, respectively. Recalling the results of these examples, we note that for X = 2,Eqs. (18), (19) have only the trivial solution y = 0, while for X = 1, the problem
(18), (19) has other, nontrivial, solutions. By extension of the terminology associated with Eq. (17) the values of X for which nontrivial solutions of (18), (19) occur are called eigenvalues and the nontrivial solutions themselves are called eigenfunctions. Restating the results of Examples 3 and 4, we have found that X = 1 is an eigenvalue of the problem (18), (19) and that X = 2 is not. Further, any nonzero multiple of sin x is an eigenfunction corresponding to the eigenvalue X = 1.
Let us now turn to the problem of finding other eigenvalues and eigenfunctions of the problem (18), (19). We need to consider separately the cases X > 0, X = 0, and X < 0, since the form of the solution of Eq. (18) is different in each of these cases. Suppose first that X > 0. To avoid the frequent appearance of radical signs, it is convenient to let X = j2 and to rewrite Eq. (18) as
y" + J2y = 0. (20)
The characteristic polynomial equation for Eq. (20) is r2 + j2 = 0 with roots r = ħi j, so the general solution is
y = c1 cos jx + c2sin jx. (21)
Note that j is nonzero (since X > 0) and there is no loss of generality if we also assume that j is positive. The first boundary condition requires that c1 = 0 and then the second boundary condition reduces to
c2 sin jn = 0. (22)
We are seeking nontrivial solutions so we must require that c2 = 0. Consequently, sin jn must be zero and our task is to choose j so that this will occur. We know that the sine function has the value zero at every integer multiple of n so we can choose j to be any (positive) integer. The corresponding values of X are the squares of the positive integers, so we have determined that
X1 = 1, X2 = 4, X3 = 9,..., Xn = n2,... (23)
are eigenvalues of the problem (18), (19). The eigenfunctions are given by Eq. (21) with c1 = 0, so they are just multiples of the functions sin nx for n = 1, 2, 3,.... Observe that the constant c2 in Eq. (21) is never determined, so eigenfunctions are determined only up to an arbitrary multiplicative constant [just as are the eigenvectors of the matrix problem (17)]. We will usually choose the multiplicative constant to be 1 and write the eigenfunctions as