By Carl M. Bender
ISBN-10: 007004452X
ISBN-13: 9780070044524
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Y. 5). ". 4). 8 (a) Show that the initial-value problem yy' I, y(O) = 0 is not well posed. Find two solutions. (b) Is d'y/dx' - 4/" + 3y" + 41' 4y = 0 (y(O) 1, y( + 00) OJ well posed? 9 Assume that y,(x) is a solution of y" + p ,(x)y' + Po(x)y = 0 From the definition of the Wronskian and Abel's formula, derive an expression for another linearly independent solution. Observe that the second solution contains y,(x) as a factor. 7~ (E) (I) (TE) (E) UO Use reduction of order to obtain the repeated root solution of y'" 3y" + 31' - y = O.
Thus, a, = If,'-·". +, Example 7 an a;"/a,, This equation can also be solved by taking logarithms of both sides: In a.. 2 2 In a•• , + In a, = O. Thus. the second discrete derivative of In a, is zero. Two coosecutive summations give In a, c, n + ", so that a, e"'·". where c, and c, are arbitrary summation constants. +, Example 8 a. = 2a; - L This equation is soluble because the transcendental functions cos x and cosh x satisfy the functional relation I(2x) 2[f(xJ]2 L If Ia, I $ I, we substitute a, cos e, and if la,1 2.
Is the Hermite polynomial of degree n [He o (x) '" 1, He , (x) x. He, (x) = x' - 1, .. J. It will be shown in Example 9 of Sec. 8 that E = n +! 4). 4b) will be shown (see Example 4 of Sec. 5) to be equivalent to the seemingly weaker inhomogeneous constraints y(x) bounded as x - ± 00. 4a) bounded ness of y(x) as Ixi- 00 implies that y_O as lxi-CD Example 4 Eigenvalue problem having transcendental eigenvalues. Consider the eigenvalue problem y" + (E - x)y 0 (0 < x < co), y(O)=O, y(oo) = O. 6) It will be shown in Example 5 of Sec.
Advanced mathematical methods for scientists and engineers by Carl M. Bender
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