28 through 36 indicate a way of solving nonhomogeneous

Chapter 11, Problem 35

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28 through 36 indicate a way of solving nonhomogeneous boundary value problems that is analogous to using the inverse matrix for a system of linear algebraic equations. The Greens function plays a part similar to the inverse of the matrix of coefficients. This method leads to solutions expressed as definite integrals rather than as infinite series. Except in 35, we will assume that = 0 for simplicityConsider the boundary value problemL[y] = [p(x)y]+ q(x)y = r(x)y + f(x), (i)1y(0) + 2y(0) = 0, 1y(1) + 2y(1) = 0. (ii)According to the text, the solution y = (x) is given by Eq. (13), where cn is defined byEq. (9), provided that is not an eigenvalue of the corresponding homogeneous problem.In this case it can also be shown that the solution is given by a Greens function integralof the formy = (x) =10G(x,s,)f(s) ds. (iii)Note that in this problem the Greens function also depends on the parameter .(a) Show that if these two expressions for (x) are to be equivalent, thenG(x,s,) = i=1i(x)i(s)i , (iv)where i and i are the eigenvalues and eigenfunctions, respectively, of Eqs. (3), (2) of thetext. Again we see from Eq. (iv) that cannot be equal to any eigenvalue i.(b) Derive Eq. (iv) directly by assuming that G(x,s,) has the eigenfunction expansionG(x,s,) = i=1ai(x,)i(s). (v)Determine ai(x,) by multiplying Eq. (v) by r(s)j(s) and integrating with respect to sfrom s = 0 to s = 1.Hint: Show first that i and i satisfy the equationi(x) = (i ) 10G(x,s,)r(s)i(s) ds. (vi)