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By Carl M. Bender

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Example 2 Wrorukia" for two functions. Two functions a. and b, are linearly dependent for ". ,;;" ,;; '" if b, is a constant multiple of a,. To see whether the Wronskian provides a test of the linear dependence of two functions we compute w" Suppose first that b. is indeed a constant multiple of il. in some domain: h. = CIl, (n, n';; "1)' Then the Wronskian vanishes in this domain: w" Il. Il. :s; n,~ Now conn';; ",. Then, assuming that a, l' Oand h, l' 0, sider the converse. Suppose that W, a for we have a,,+tia.

1) reduces this equation to one that is first order: uu' u yu. This equation implies that either u 0 so that y(x) c 3 , a constant, or et~e u'(y) = y + I, whose solution is u(y) y'/2 + y + C" If we now recall that u(y) Y'(t) and that x 2', then we must finally solve the separable equation Y'(t) = y' /2 + y + C I for y(x). The final solution is given in (I. LlOHSee also Prob. 3) leaves the original differential equation unchanged. Example 3 Scale-invariant equations. 2 is scale invariant under the transformation x -+ ax, Y -+ a- 3 y.

Note that we do not Integrate from -w to +w. Why? nG(x, a)J(a) do Example 5 Solution oj a boundary-value problem by Green's Junctions. The Green's function for the boundary-value problem y"- y =J(x)[y(±x) 0] is defined by the equations a2 GjiJxi G(x, a) ,s(x - oj, (I) = O. I . IJ(a)da. Reduction of Order For the sake of completeness, it is important to state that reduction of order reduces the order of inhomogeneous as well as homogeneous equations_ Thus, since all first-order linear equations are soluble, reduction of order is especially useful for second-order linear equations.

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