homogeneous system of ODE's with non-constant coefficients

I'm using Maxima 5.12.0 to try to solve the following homogeneous
linear system of ODE's with non-constant coefficients:

eqn_1: 'diff(x[1](t), t, 1) =   x[1](t)             + t^2*x[3](t)$
eqn_2: 'diff(x[2](t), t, 1) = 2*x[1](t) + t*x[2](t) +   4*x[3](t)$
eqn_3: 'diff(x[3](t), t, 1) =                           3*x[3](t)$

but it seems like Maxima isn't able to solve it directly:

(%i5)        desolve([eqn_1, eqn_2, eqn_3], [x (t), x (t), x (t)])
                                              1      2      3
`algsys' cannot solve - system too complicated.
`desolve' can't handle this case.
 -- an error.  To debug this try debugmode(true);


I know I can solve this by first finding x3(t) out of the third equation,
then substituting this in the first and obtaining x1(t).  Finally substitute
x1(t) and x3(t) into the second equation to obtain x2(t).  However, more
out of curiosity, I was wondering if there is an alternative way to declare
this problem to Maxima so that it is able to solve it.... because Maple
seems to do the job :-(

Thanks,
Bart

-- 
	"Share what you know.  Learn what you don't."