O15-C907
> restart:
> f:=x->exp(x)*cos(x)-exp(2*x);

                  f := x -> exp(x) cos(x) - exp(2 x)

> eq:=x->(D@@3)(f)(x)+a*(D@@2)(f)(x)+b*(D@@1)(f)(x)+c*f(x):eq(x);

  -2 exp(x) sin(x) - 2 exp(x) cos(x) - 8 exp(2 x)

         + a (-2 exp(x) sin(x) - 4 exp(2 x))

         + b (exp(x) cos(x) - exp(x) sin(x) - 2 exp(2 x))

         + c (exp(x) cos(x) - exp(2 x))

> s:=solve({eq(0),eq(1),eq(2)},{a,b,c});

                     s := {a = -4, c = -4, b = 6}

> subs(s,eq(x));

                                  0

> p:=x^3-4*x^2+6*x-4;

                             3      2
                       p := x  - 4 x  + 6 x - 4

> rac:=solve(p);

                        rac := 2, 1 + I, 1 - I

> reste:=proc(n) rem(x^n,p,x) end;

                 reste := proc(n) rem(x^n, p, x) end

> sort([reste(4),reste(5)]);

                    2                          2
               [20 x  - 44 x + 40, -20 x + 10 x  + 16]

> # R_n se décompose dans une base quasi-Lagrange : u_n=R_n(c)/(c-a)/(c-b) et R_n(c)=c^n
> eq1:=x->a*(D@@2)(f)(x)+b*(D@@1)(f)(x)+c*f(x):
> s:=solve({eq1(0),eq1(1),eq1(2)},{a,b,c});# Donc (f,f',f") est libre

                      s := {a = 0, c = 0, b = 0}

> # On résout (D-aId)°(D-bId)°(D-cId)(y)=0
> a:=rac[1]:b:=rac[2]:c:=rac[3]:
> ed1:=diff(y(x),x)-a*y(x):soled1:=dsolve(ed1,y(x)):z1:=subs(soled1,y(x));

                          z1 := _C1 exp(2 x)

> ed2:=diff(y(x),x)-b*y(x)-z1:soled2:=dsolve(ed2,y(x)):z2:=subs(soled2,y(x));

  z2 := 1/2 _C1 exp((1 - I) x + (1 + I) x)

         + 1/2 I _C1 exp((1 - I) x + (1 + I) x) + _C2 exp((1 + I) x)

> ed3:=diff(y(x),x)-c*y(x)-z2:soled3:=dsolve(ed3,y(x)):z3:=subs(soled3,y(x));

  z3 := 1/2 _C1 exp((1 - I) x + (1 + I) x)

         - 1/2 I _C2 exp(2 I x + (1 - I) x) + _C3 exp((1 - I) x)

>