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

(First math examination for the M' section, 1986). Duration: 4h

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

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

Download files in pdf or LaTeX formats

Sketch of the solution

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

y
 = 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.

I

I.1 How a solution of (E), the graph of which is included in the half-plane defined by the inequality x < 0, can be deduced from a solution, the graph of which is included in the half-plane defined by x > 0 ? (Solution)

I.2 Let l be a given real number, different from 0, and j a maximal solution of (E) defined on an interval which contains the number 1. How can we deduce from j the maximal solution j l such that jl (l)=lj(1) and jl (l)=j(1) ? (Solution)

I.3 How the solutions of (E), the graph of which are included in one of the quarter-of-plane defined by the couple of inequalities (x < 0,y > 0) , (x < 0,y < 0) , (x > 0,y > 0), can be deduced from the solutions, the graph of which is included in one of the quarter-of-plane defined by (x > 0,y > 0) ? (Solution)

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 S, defined on the open interval I=]a,b[ ]0,+[.

II.1.a For all x I, using j(x), give the expression of the derivative (x2j(x)) and specify its sign. (Solution)

II.1.b Can the function j have a minimum value on I? (Solution)

II.1.c Show that j is monotonous on I or a sub-interval [a,b[ that has the same right edge as I. (Solution)

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

II.2.a Justify the inequality j1(x0) j2(x0). (Solution)

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 ]xo,b[ such that j 1(x)=j2(x)

Denoting by x1 the inf value of the real numbers x that obey H1, express for all x [x0,x1] the difference x2j 2(x)-x2j1(x) using x02[j2(x0)- j1(x0)] and an integral which involves j1 and j2. Deduce from that a comparison between j1(x) and j2(x) then between j1(x) and j2(x). Is the H1 hypothesis acceptable? (Solution)

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,+[ with a 0
and we denote by g a point in ]a,+[.

II.3.a Compare x2j(x) and g2j(g) for all x g and conclude that j is bounded on [g,+[. (Solution)

II.3.b When x goes to +, tudy the behavior of x2j(x) and conclude that j tends to -. (Solution)

II.3.c Is H2 an acceptable hypothesis? (Solution)

II.4 Let j be a solution in S.

II.4.a From the previous question, conclude that the definition interval ]a,b[ is bounded. What is the limit of j when x tends to b? (Solution)

II.4.b Assuming that j has a finite limit l when x tends to b, find an upper bound for j in a neighborhood of b and compute from that an upper bound for x2j(x) that is incompatible with the convergence of j to l. What is the limit of j when x tends to b? (Solution)

II.5 In this question and the following, we assume that there exists a solution j 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.

II.5.a What is the value of y(0)? What is the sign of y(c)? (Solution)

II.5.b Let y1 S, defined on ]a1,b1[ with a1 < c < b1 and such that y1(c) = y(c) and y1(c) < y(c). Check that a1=0. When x tends to 0, get the sign of the limit of x2y1(x) -x2y(x) and get from that the limit of y1, then justify the existence of a strictly non-negative lower bound for xy1(x). What is the limit of y1(x) when x tends to 0? (Solution)

II.5.c Get the sign of (xy1(x))" and establish the existence of an upper bound for xj1(x) that is independent of x ]0,b1[.
Has the product xj1(x) a finite,non-vanishing limit when x tends to 0? (
Solution)

II.5.d Let y2 S, defined on ]a2,b2[ with a2 < c < b2 and such that y2(c) = y(c) and y2(c) > y(c). What is the sign of x2y2(x) -x2y(x) on ]a2,c[? Find a lower bound for x2y 2(x) in a neighborhood of a2 and conclude that a2 cannot be 0. What are the limits of y2 and of y2 when x tends to a2? (Solution)

II.6 Assuming that the hypothesis of question II.5 are valid for c=1, draw onn one shema the graphs of the different kinds of solution in S that contain the point (1,y(1)). (Solution)

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.a For all x [0,1], compute using integrals involving y the values of x2y(x)- y(1) , of xy (x)+y(x) -y (1)-h and of y(1). (Solution)

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)=
1

0 
(t-t2)f(t)dt

and, if x ]0,1]


T(f)(x)=
 1

x
-1

x

0 
t2f(t) dt +
1

x 
(t-t2)f(t)

Check the equality y = h+T(1/y). (Solution)

III.2.a Compute the derivatives T(f)(x) and T(f)(x). Find their values for x=0? Check that T(f) is of class C2 on the closed interval [0,1]. (Solution)

III.2.b Show that if f is non-negative on [0,1], then T(f) is strictly non-negative on [0,1[, except for a particular choice for f that you will specify. What is the value of T(f)(1)? (Solution)

III.2.c Check the linearity of f -> T(f) defined from the set E of continuous real functions on [0,1] into the set F of real functions that are of class C2 on [0,1], and check that if one denotes ||f|| = sup |f(x)| for 0 x 1, then for all f E, one has ||T(f)||  1/6 ||f||. (Solution)

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

gn+1 = h+T( 1

gn
)

III.3.a Check that gn lies in F for all n N. (Solution)

III.3.b Show that for every integer p 0 one has g2p g2p+1, and that the sub-sequences (g2p) and (g2p+1) are increasing for one, decreasing for the other. (Solution)

III.3.c Conclude that (g2p) converges on [0,1] to a real function g, and that (g2p+1) converges on [0,1] to a real function G. (Solution)

III.4.a Find a real number that is an upper bound of the absolute value of the derivative gn for all n. (Solution)

III.4.b Conclude that for each e > 0 one can find a partition of [0,1] made of intervals, each of the same length, such that for all n, the image by gn of each of these intervals is an interval, the length of which is less than e. (Solution)

III.4.c Show that the convergence of (g2p) and (g2p+1) are both uniform on [0,1]. (Solution)

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

Are the functions g and G in F? (Solution)

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

III.5.a Give the values of u(0), u(1) and u(0). (Solution)

III.5.b Give the expression of u in terms of u(x), g(x), G(x). (Solution)

III.5.c What is the function u? How can we obtain the solution y of (E) that is looked for in this part III? (Solution)

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)