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Unformatted text preview: 'r-idIi",it,ELEMENTARY DIFFERENTIALEQUATIONS ANDBOUNDARY.. V ALUEPROBLEMSo.al19tm)yht)yisWilliam E. BoyceandRichard C. DiPrimaRensselaer Polytechnic InstituteSecond Edition~ -.1-*2.11 THE EXISTENCE AND UNIQUENESS THEOREMIn this section we wil discuss the proof of Theorem 2.2, the fundamentalexistence and uniqueness theorem for first order initial value problems. Thistheorem states that under certain conditions on ¡(x, y), the iIiitial valueproblemy,igofy' = ¡(x, y),y(xo) = Yo(1 a)(lb)ifIi,ihas a unique solution in some interval containing the point Xo.In some cases (for example, if the differential equation is linear) theexistence of a solution of the initial value problem (1) can be establisheddirectly by actually solving the problem and exhibiting a formula for theiu l'irst uraer umeremial bquationssolution. However, in general this approach is not feasible because there isno method of solving Eq. (la), which applies in all cases. Therefore, for thegeneral case it is necessary to adopt an indirect approach that establishes theexistence of a solution of Eqs. (1) but usually does not provide a means offinding it. The heart of this method is the construction of a sequence offunctions which converges to a limit function that satisfies the initial valueproblem, although the members of the sequence individually do not. As a.rule, it is impossible to compute explicitly more than a few members of thesequence; therefore the limit function can be found explicitly only inrare cases, and hence this method does not provide a useful way of actuallysolving initial value problems. Nevertheless, under the restrictions on¡(x, y) stated in Theorem 2.2, it is possible to show that the sequence inquestion converges and that the limit function has the desired properties.The argument is fairly intricate and depends, in part, on techniques andresults that are usually encountered for the first time in a course on advancedcalculus. Consequently, we wil not go into all of the details of the proofhere; we wil, however, indicate its main features and point out some of thediffculties involved.First of all, we note that it is suffcient to consider the problem in whichthe initial point (xo, Yo) is the origin; that is, the problemy' = j(x, y),y(O) = o.(2a)(2b)If some other initial point is given, then we can always make a preliminarychange of variables, corresponding to a translation of the coordinate axes,which wil take the given point (xo, Yo) into the origin. Specifically, weintroduce new dependent and independent variables wand s, respectively,which are defined by the equationsw = y - Yo,(3)s = x - Xo;see also Figure 2.11. Thinking of was a function of s, we have by the chainruledw = dw dx = ..-E (y _ Yo) dx = dy ....
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