This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1906 Excerpt: ...number of ways, by imposing any other n--i conditions we please. In actual practice we shall impose these conditions in such a way as to simplify our work as much as possible. The method will be carried out in the case of an equation of the third order. (The argument will readily be seen to apply to any order.) Let the ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1906 Excerpt: ...number of ways, by imposing any other n--i conditions we please. In actual practice we shall impose these conditions in such a way as to simplify our work as much as possible. The method will be carried out in the case of an equation of the third order. (The argument will readily be seen to apply to any order.) Let the equation be (i). (k"ZP + IF + k2D + yX, and let the complementary function be ( 2) y = cjf, + c2em + Ctff. We shall try to find cu c2, cit such that (2) shall be a solution of (1). This still allows us to impose two conditions upon cu c2, c3. Differentiating (2), we get Dy = + mgcifV + miclem + +. + em dx dx dx We shall now use one of the two conditions at our disposal by letting (.z) (-,"--x + "vf + em' = o; dx dx dx so that we have (4) Dy--mxcxem, + mgm? + msc3em. If any of the roots of the auxiliary equation are repeated or imaginary, the resulting change in the form of the complementary function causes no difference in the process. thus using the second condition still at our disposal, so that (6) I?y = mfc? + mic./"? + mfcff. Differentiating again, we get (7) LPy = m?cxen? + mictfmf + SV" + m?em? 3 + /2Vv dx dx dx Substituting (2), (4), (6), (7), in (1), and remembering that with fi, c2, c3 constant, (2) is the complementary function, we have (8) W P + frf + W-" = X..dx dx dx Equations (3), (5), (8) are three linear equations sufficient to determine--------, and by quadratures cu '2, cs will be found such dx dx dx that (2) will be a solution of (1), the constants of integration giving us again the complementary function. The method of variation of parameters applies to all linear equations, whether the coefficients are constants or not. (Thus, see Ex. 4, 53.) As an ill...
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