@Preben Alsholm 

thank u very very very much u helped me very much ;)

can i use this as explanation 

restart;
with(inttrans);
k := diff(u(t, x), t,t) = diff(u(t, x), x,x); 
bc := u(0, x) = 1/(1+x^2);
v:=D[1](u)(0,x)=0;

eq1:=map(fourier,k,x,w); #we integrat here both side of equation
eq2:=subs(fourier(u(t, x), x, w)=g(t),eq1); # sequential substitution where we put the equation depend on g(t)
inits:=g(0)=fourier(rhs(bc),x,w), D(g)(0)=0;#what it is doing here??
dsolve({eq2,inits}); #solving with the sequential substution and the inits
invfourier(rhs(%),w,x) assuming t::real; #resubstituion the result of solving 

is it right??

@Preben Alsholm 

thank u my next question is how i must solve it special with fourier transformation rezpect to x ? i thought that i use the first step like before 

k := diff(u(t, x), t,t) = diff(u(t, x), x,x); 
bc := u(0, x) = 1/(1+x^2);
v:=D[1](u)(0,x)=0;

F(w) = (int(k*e^(-I*w*x), x = -infinity .. infinity))/sqrt(2*Pi);

but it doesent work how i solve it special with fourier transformation?

@Preben Alsholm 

hey first of all thank u very much for ur help and i wish u happy new year ;)

hey Preben Alsholm can u tell me how can i PLOT it

i thought something like plot(res,u(t,x)=-10..10,x=-10..10);

but it didn`t work can u help me please?

@Carl Love 

i have already check it by hand both are the same sorry for the confusen :D

@Preben Alsholm 

hey first of all thank u very much for ur help and i wish u happy new year ;)

hey Preben Alsholm can u tell me how can i PLOT it

i thought something like plot(res,u(t,x)=-10..10,x=-10..10);

but it didn`t work can u help me please 

@Preben Alsholm 

first of all thank u very much for the quick answer but can u explain me why i need to use D and not diff

and from step to step what did it do and why:

eq1:=subs(t=0,bc,res);
diff(res,t);
convert(%,D);
eq2:=subs(t=0,v,%);
eq3:=map(int,eq2,x)+(C=0);
solve({eq1,eq3},{_F1(x),_F2(x)});
_F1,_F2:=op(unapply~(rhs~(%),x));
res;
sol:=normal(%);

maybe it´s a stupid question but i am very new in programming in maple and i want to understand it what this programm code do step to step because next time i want to programm it on my self ;)

and thank u once more :D