O15-C909
> restart:
> ed:=diff(x(t),t,t)-t*x(t);

                            / 2      \
                            |d       |
                      ed := |--- x(t)| - t x(t)
                            |  2     |
                            \dt      /

> dsolve(ed,x(t));

                 x(t) = _C1 AiryBi(t) + _C2 AiryAi(t)

> sol1:=dsolve({ed,x(0)=1,D(x)(0)=0},x(t));sol2:=dsolve({ed,x(0)=0,D(x)(0)=1},x(t));# Cauchy-Lipschitz linéaire d'ordre 2

                                       (1/6)
  sol1 := x(t) = GAMMA(2/3) AiryAi(1) 3

                                                      /
        (3 AiryAi(1) + sqrt(3) AiryBi(1)) AiryBi(t)  /  (
                                                    /

                   2            2                (1/6)
        3 AiryAi(1)  - AiryBi(1) ) - GAMMA(2/3) 3

                                                                /
        (3 AiryAi(1) + sqrt(3) AiryBi(1)) AiryBi(1) AiryAi(t)  /  (
                                                              /

                   2            2
        3 AiryAi(1)  - AiryBi(1) )


                   (sqrt(3) AiryAi(1) + AiryBi(1)) AiryBi(t)
  sol2 := x(t) = - -----------------------------------------
                                      2            2
                           3 AiryAi(1)  - AiryBi(1)

           (sqrt(3) AiryAi(1) + AiryBi(1)) sqrt(3) AiryAi(t)
         + -------------------------------------------------
                                  2            2
                       3 AiryAi(1)  - AiryBi(1)

> courb1:=subs(sol1,x(t));courb2:=subs(sol2,x(t));

                                  (1/6)
  courb1 := GAMMA(2/3) AiryAi(1) 3

                                                      /
        (3 AiryAi(1) + sqrt(3) AiryBi(1)) AiryBi(t)  /  (
                                                    /

                   2            2                (1/6)
        3 AiryAi(1)  - AiryBi(1) ) - GAMMA(2/3) 3

                                                                /
        (3 AiryAi(1) + sqrt(3) AiryBi(1)) AiryBi(1) AiryAi(t)  /  (
                                                              /

                   2            2
        3 AiryAi(1)  - AiryBi(1) )


              (sqrt(3) AiryAi(1) + AiryBi(1)) AiryBi(t)
  courb2 := - -----------------------------------------
                                 2            2
                      3 AiryAi(1)  - AiryBi(1)

           (sqrt(3) AiryAi(1) + AiryBi(1)) sqrt(3) AiryAi(t)
         + -------------------------------------------------
                                  2            2
                       3 AiryAi(1)  - AiryBi(1)

> plot([courb1,courb2],t=-8..2.5);

> # Les solutions ayant un DSE forment un ev de dimension 2 (a_0 et a_1 sont quelconques, a_2=0) donc on a toutes les solutions
> f:=t->int(cos(u*t-u^3/3),u=0..Pi)/Pi;

                              Pi
                             /
                            |                   3
                            |    cos(u t - 1/3 u ) du
                            |
                           /
                             0
                 f := t -> --------------------------
                                       Pi

> plot(f(t),t=-8..2.5);

> # f est solution (particulière) de y"+ty=1/Pi.sin(Pi.t-Pi^3/3) ; t->x_1(-t) et t->x_2(-t) sont deux solutions indépendantes de l'EDO homogène associée
> # t->f(t)+x_1(-t)+x_2(-t) est donc la solution générale de y"+ty=1/Pi.sin(Pi.t-Pi^3/3)