Notes from the author (updated 20030917):
Download files in pdf or LaTeX formats
Your comments are welcome ...
eric.chopin@wanadoo.fr
(Note, this email is encrypted so that spammers cannot catch it using robots, see how...
BACK TO THE HOME PAGE  visitors: from 08/28/2003
The goal of the project is to study maximal, real solutions of the differential equation:

I.1 How a solution of (E), the graph of which is included in the halfplane defined by the inequality x < 0, can be deduced from a solution, the graph of which is included in the halfplane 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 j_{l} (l)=lj(1) and j¢_{l} (l)=j¢(1) ? (Solution)
I.3 How the solutions of (E), the graph of which are included in one of the quarterofplane 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 quarterofplane defined by (x > 0,y > 0) ? (Solution)
II.1 Let j Î S, defined on the open interval I=]a,b[ Ì ]0,+¥[.
II.1.a For all x Î I, using j(x), give the expression of the derivative (x^{2}j¢(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 subinterval [a¢,b[ that has the same right edge as I. (Solution)
II.2.a Justify the inequality j_{1}¢(x_{0}) ¹ j_{2}¢(x_{0}). (Solution)
Denoting by x_{1} the inf value of the real numbers x that obey H_{1}, express for all x Î [x_{0},x_{1}] the difference x^{2}j _{2}¢(x)x^{2}j_{1}¢(x) using x_{0}^{2}[j_{2}¢(x_{0}) j_{1}¢(x_{0})] and an integral which involves j_{1} and j_{2}. Deduce from that a comparison between j_{1}¢(x) and j_{2}¢(x) then between j_{1}(x) and j_{2}(x). Is the H_{1} hypothesis acceptable? (Solution)
II.3.a Compare x^{2}j¢(x) and g^{2}j¢(g) for all x ³ g and conclude that j is bounded on [g,+¥[. (Solution)
II.3.b When x goes to +¥, tudy the behavior of x^{2}j¢(x) and conclude that j¢ tends to ¥. (Solution)
II.3.c Is H_{2} 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 x^{2}j¢(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.a What is the value of y¢(0)? What is the sign of y¢(c)? (Solution)
II.5.b Let y_{1} Î S, defined on ]a_{1},b_{1}[ with a_{1} < c < b_{1} and such that y_{1}(c) = y(c) and y_{1}¢(c) < y¢(c). Check that a_{1}=0. When x tends to 0, get the sign of the limit of x^{2}y_{1}¢(x) x^{2}y¢(x) and get from that the limit of y_{1}¢, then justify the existence of a strictly nonnegative lower bound for xy_{1}(x). What is the limit of y_{1}(x) when x tends to 0? (Solution)
II.5.c Get the sign of (xy_{1}(x))"
and establish the existence of an upper bound
for xj_{1}(x) that is independent of x Î
]0,b_{1}[.
Has the product xj_{1}(x)
a finite,nonvanishing limit when x tends to 0? (Solution)
II.5.d Let y_{2} Î S, defined on ]a_{2},b_{2}[ with a_{2} < c < b_{2} and such that y_{2}(c) = y(c) and y_{2}¢(c) > y¢(c). What is the sign of x^{2}y_{2}¢(x) x^{2}y¢(x) on ]a_{2},c[? Find a lower bound for x^{2}y _{2}¢(x) in a neighborhood of a_{2} and conclude that a_{2} cannot be 0. What are the limits of y_{2} and of y_{2}¢ when x tends to a_{2}? (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.1.a For all x Î [0,1], compute using integrals involving y the values of x^{2}y¢(x) y¢(1) , of xy ¢(x)+y(x) y ¢(1)h and of y¢(1). (Solution)


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 C^{2} on the closed interval [0,1]. (Solution)
III.2.b Show that if f is nonnegative on [0,1], then T(f) is strictly nonnegative 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 C^{2} on [0,1], and check that if one denotes f = sup f(x) for 0 £ x £ 1, then for all f Î E, one has T(f) £ 1/6 f. (Solution)
III.3 We define by recursion a sequence of functions g_{n} by g_{0}=h and, for every integer n ³ 0,

III.3.a Check that g_{n} lies in F for all n Î N. (Solution)
III.3.b Show that for every integer p ³ 0 one has g_{2p} £ g_{2p+1}, and that the subsequences (g_{2p}) and (g_{2p+1}) are increasing for one, decreasing for the other. (Solution)
III.3.c Conclude that (g_{2p}) converges on [0,1] to a real function g, and that (g_{2p+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 g_{n}¢ 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 g_{n} 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 (g_{2p}) and (g_{2p+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)