From the help page dsolve, numeric:
"a procedure is returned that can be used to obtain solution values if given the value of the independent variable"
This means that X(t) is not a sort of "algebraic formula" with t's in it.
If t is a numeric value: X(t) calculates a numeric value for X:

> X(0.1);

returns 0.9856922284 or so, but if t has no value,

 
> X(t);

stays unevaluated.
If you want to know what X is:

> showstat(X);

Assuming that u is known, and defined as

u :=  x ->...

you can do:

s := dsolve( {DE,BC }, x(t), numeric, output = listprocedure );
U := u @ subs(s,x(t));
plot( U, 0..1 );

There is an unassigned parameter l in the ODE.

If you change the command in:

sol := dsolve({bc, eqn}, numeric, parameters=[l]);

you get the errormessage:

Error, (in dsolve/numeric) cannot numerically solve a parametric boundary value problem

Additionally: there are 5 ic/bc's and that results in an inconsistent system of equations if you try to solve for the constants in the general solution.
Numerical solutions can be found for 4 initial conditions.

Of course this is possible, but no very easy.

Use the option output=list:

with(LinearAlgebra):
A := <a11,-a11,a13; 0,0,0; 0,0,0>;
eig := Eigenvectors(A, output=list);

s := sort( eig, (x,y)->type(x[1],symbol) ); # if you want the nonzero element(s) first
s := sort( eig, (x,y)->x[1]=0 ); # if yoy want the zero(s)  first
V := Matrix( map( op, s[1..2,3] ) ); # the matrix of eigenvectors