> restart;
O14-C100

> # g et h sont paires ; h'(x)=-x/2 g(x) ; (-ix g(x) /2)'=i(g(x)-h(x))
> g:=int(cos(t*x)/(1+t^2),t=- infinity..+ infinity);

                      g := -signum(x) Pi sinh(x)

> h:=int(cos(t*x)/(1+t^2)^2,t=-infinity..infinity);

     h := - 1/2 signum(x) Pi sinh(x) + 1/2 x signum(x) Pi cosh(x)

> int(1/(1+t^2),t=-infinity..infinity);int(1/(1+t^2)^2,t=-infinity..infinity);

                                  Pi


                                1/2 Pi


> # g vérifie g"=g , g(0)=Pi , g est paire , lim g=0 en + infinity
> g2:=Pi*exp(-abs(x));

                         g2 := Pi exp(-| x |)

> eq2:={diff(z(x),x)+x*g2/2,z(0)=Pi/2};

                              /d      \
       eq2 := {z(0) = 1/2 Pi, |-- z(x)| + 1/2 x Pi exp(-| x |)}
                              \dx     /

> sol2:=dsolve(eq2,z(x)):h2:=subs(sol2,z(x));

                       2                       2
                   Pi x  exp(-| x |)       Pi x  exp(-| x |)
         h2 := 1/2 ----------------- + 1/2 -----------------
                         | x |                       2
                                                | x |

> plot([g2,h2],x=-10..10,legend=["g","h"],scaling=constrained);
>