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19. If W (f, g) is the Wronskian of f andg, and if u = 2 f g,v = f + 2g, find the Wronskian W(u,v) of u and v in terms of W(f, g).
20. If the Wronskian of f and g is t cos t sin t, and if u = f + 3g, v = f g, find the Wronskian of u and v.
In each of Problems 21 and 22 find the fundamental set of solutions specified by Theorem 3.2.5 for the given differential equation and initial point.
21. y" + y 2y = 0, t0 = 0 22. y" + 4y' + 3y = 0, ^ = 1
In each of Problems 23 through 26 verify that the functions y1 and y2 are solutions of the given differential equation. Do they constitute a fundamental set of solutions?
23. y" + 4y = 0; y^t) = cos2t, y2(t) = sin2t
24. y" 2y' + y = 0; y1 (t) = et, y2(t) = tet
25. x2y" x(x + 2)y; + (x + 2)y = 0, x > 0; y1(x) = x, y2(x) = xex
26. (1 x cotx)y" xy + y = 0, 0 < x < n; y1(x) = x, y2(x) = sinx
27. Exact Equations. The equation P(x)y" + Q(x)y! + R(x)y = 0 is said to be exact if it
can be written in the form [P (x)y'] + [ f (x)y] = 0, where f (x) is to be determined in terms of P(x), Q(x), and R(x). The latter equation can be integrated once immediately, resulting in a first order linear equation for y that can be solved as in Section 2.1. By equating the coefficients of the preceding equations and then eliminating f (x), showthat a necessary condition for exactness is P"(x) Q'(x) + R(x) = 0. It can be shown that this is also a sufficient condition.
In each of Problems 28 through 31 use the result of Problem 27 to determine whether the given equation is exact. If so, solve the equation.
28. y" + xy + y = 0 29. y" + 3x2y; + xy = 0
30. xy" (cos x)y'+ (sinx)y = 0, x > 0 31. x2y" + xy' y = 0, x > 0
32. The Adjoint Equation. If a second order linear homogeneous equation is not exact, it can be made exact by multiplying by an appropriate integrating factor fi(x). Thus we require that fi(x) be such that fi(x) P (x) y" + fi(x) Q (x) y + p(x) R(x )y = 0 can be written in the form [^(x)P(x)y']' + [ f (x)y] = 0. By equating coefficients in these two equations and eliminating f (x), show that the function p, must satisfy
P p" + (2 P' Q)p + (P" Q' + R)p = 0.
This equation is known as the adjoint of the original equation and is important in the advanced theory of differential equations. In general, the problem of solving the adjoint differential equation is as difficult as that of solving the original equation, so only occasionally is it possible to find an integrating factor for a second order equation.
3.3 Linear Independence and the Wronskian
In each of Problems 33 through 35 use the result of Problem 32 to find the adjoint of the given
33. x2y" + xy' + (x2 v2)y = 0, Bessels equation
34. (1 x2)y" 2xy' + a(a + 1)y = 0, Legendres equation
35. y" xy = 0, Airys equation
36. For the second order linear equation P (x )y" + Q (x )y' + R(x )y = 0, show that the adjoint of the adjoint equation is the original equation.
37. A second order linear equation P(x)y" + Q(x)y' + R(x)y = 0 is said to be self-adjoint if its adjoint is the same as the original equation. Show that a necessary condition for this equation to be self-adjoint is that P' (x) = Q (x). Determine whether each of the equations in Problems 33 through 35 is self-adjoint.
3.3 Linear Independence and the Wronskian
The representation of the general solution of a second order linear homogeneous differential equation as a linear combination of two solutions whose Wronskian is not zero is intimately related to the concept of linear independence of two functions. This is a very important algebraic idea and has significance far beyond the present context; we briefly discuss it in this section.
We will refer to the following basic property of systems of linear homogeneous algebraic equations. Consider the two-by-two system
a11 x1 + a12x2 = 0 (1)
a21 x1 + a22x2 = 0
and let A = a11a22 a12a21 be the corresponding determinant of coefficients. Then x = 0, y = 0 is the only solution of the system (1) if and only if A = 0. Further, the system (1) has nonzero solutions if and only if A = 0.
Two functions f and g are said to be linearly dependent on an interval I if there exist two constants k1 and k2, not both zero, such that
k1 f (t) + k2 g(t) = 0 (2)
for all t in I. The functions f and g are said to be linearly independent on an interval I if they are not linearly dependent; that is, Eq. (2) holds for all t in I only if k1 = k2 = 0. In Section 4.1 these definitions are extended to an arbitrary number of
functions. Although it may be difficult to determine whether a large set of functions is
linearly independent or linearly dependent, it is usually easy to answer this question for a set of only two functions: they are linearly dependent if they are proportional to each other, and linearly independent otherwise. The following examples illustrate these definitions.
EXAMPLE Determine whether the functions sint and cos(t n/2) are linearly independent or