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35 Example of non-uniqueness For the differential equation of second order and second degree or y =f(x,Y,Y')= 1 +y y"(1 +y')= 1 the conditions of boundedness and continuity are already infringed for variable y'. (2x)3 go through the point x = 0, y = 1 with the slope y' = - 1. , if the differential equation is of the form Pu(x)Y(°)(x) = r(x) with pn(x) = 1, and if we rewrite the differential equation as a system with n-1 Y(q) = Y(q+1) (q = 1, ... , n - 1), Y(n) = f = r(x) - Z PU(x)Y("11), V=0 then the partial derivatives of f with respect to the y(q) are precisely the coefficient functionspq _ i (x) for q = 1, 2, ...

We next check whether T is a contraction operator. 56) z°(E) - OI <_ P( ) II Z_ 211. Further we can now choose p(x) to be a positive continuous function on J. For p(x)=e-"I x-xoI with a>Ls we have P(x) di J XoP( ) Hence it follows altogether that II TZ - TZII <- KIIZ - ZII with K=Ls< 1. a Therefore the operator T is indeed a contraction in the domain considered. 22. BANACH'S FIXED-POINT THEOREM AND THE EXISTENCE THEOREM FOR ORDINARY DIFFERENTIAL EQUATIONS We have now introduced the concepts needed for formulating a general fixed- point theorem, from which the existence theorem for ordinary differential equations can easily be deduced.

Fig. 15. The two-valued direction field for the differential equation y'= x± 2+x 2 + y 2 23 Fig. 16. The infinitely-many-valued direction field of the tangents to a sine curve Eliminating the parameter by using cos t;= y', sin E=1-y'2 we obtain the differential equation y - j 1 - y' 2 = xy' - y' arc cosy' . 15. NON-UNIQUENESS OF THE SOLUTION For physical applications it is important to know sufficient conditions for uniqueness of the solution of a differential equation. That it is not sufficient for the slope function y' = f(x, y) to be singlevalued and continuous to ensure uniqueness of the solution is shown by the following example.

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