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Entrance examination for admission at the Ecole Polytechnique (Paris, France)(Version francaise)

Note from UPS (Union des Professeurs de Spéciale = undergraduate-teachers association): the answer to question III.5.c is still unknown .....

Notes from the author (updated 2003-09-17):

• the solution to III.5.c was not found when the examination was given, but a partial solution to question III.5.c was found in 1987 and published in the "Revue de Mathématiques spéciales", april 1987 page 284. The solution found does not take into account the indications given in questions III.5.XXX. In fact the article solves "only" the main part of the question, i.e. how to compute the maximal solutions defined on [0,1]. However it does not prove that the sequence of function gn converges (u=0) for all h > 0.
• If we except questionm III.5.c, the test is somewhat easy and make reference to most of the arguments that are typically used in functional analysis. Candidates are therefore supposed to pay some great attention to the quality of their arguments.
• This page was initially written using LaTeX and converted into HTML with a personal tool under development, similar to TTH

(Note, this email is encrypted so that spammers cannot catch it using robots, see how...

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The goal of the project is to study maximal, real solutions of the differential equation:
 (E) xy˘˘ + 2y˘ + xy = 0

where x is a real variable. All the considered solutions of E are supposed to be real-valued.
Apart from question III.6, parts I, II, et III are independent, one from the other.

II

Let S be the set of solutions of (E), the graph of which is included in the quarter-of-plane defined by the inequalities (x > 0,y > 0), and which are maximal for this quarter-of-plane.

II.1 Let j Î S, defined on the open interval I=]a,b[ Ě ]0,+Ą[.

II.2 Let j1 and j2 be two distinct solutions lying in S, both defined at least on an interval ]a,b[ Ě ]0,+Ą[. It is assumed that there exists a real number x0 Î ]a,b[ such that j1(x0)=j2(x0).

II.2.b Suppose that j1˘(x0) < j2 ˘(x0) and we also assume that the following hypothesis is valid:
H1: there exists at least a real number x Î ]xo,b[ such that j 1(x)=j2(x)

II.3 To sketch the drawing of the graphs of the solutions in S when x increases, we now assume the following hypothesis to be valid:
H2: there exists a solution j in S and defined on ]a,+Ą[ with a ł 0
and we denote by g a point in ]a,+Ą[.

II.4 Let j be a solution in S.

II.5 In this question and the following, we assume that there exists a solution j Î S defined on an interval ]0,d[, with a finite, non-vanishing, limit in x=0, the continuous prolongation of which on [0,d[ (still denoted by y) is of class C2 on a closed interval [0,c] with 0 < c < d.

III

The main goal of this part is to establish the existence and a computing scheme of a solution y of (E) which is of class C2 on the closed interval [0,1] and the value of which at x=1 is a given y(1)=h > 0 (h being arbitrarily set). In these conditions, y˘(0)=0.

III.1.b For any given continuous function f defined on [0,1] one associate the other function T(f) defined on [0,1] by:

 T(f)(0)= óő 10 (t-t2)f(t)dt

and, if x Î ]0,1]

 T(f)(x)= ćč 1x -1 öř óő x 0 t2f(t) dt + óő 1 x (t-t2)f(t)

III.3 We define by recursion a sequence of functions gn by g0=h and, for every integer n ł 0,

 gn+1 = h+T( 1 gn )

III.4.d Conclude that the following equalities are true
g = h+T(1/G) and G = h+T(1/g)

III.5 We define on [0,1] the function u by the equality u(x)=x[G(x)-g(x)].

III.6 Sketch the drawing of the graph of a maximal solution of E that contains the point (0,1). How can we get from that the graphs of the other maximal solutions of E defined when x=0 ? (Solution)