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solution of system of differential equations

is {a(t) = 1, b(t) = exp(t), c(t) = 0}

1 = exp(t), since t = 0, is characteristic equation = (y-0)?
0 = exp(t), how to find this characteristic equation when division is zero?
exp(t) = exp(t), is characteristic equation = (y-t)?

after solved, 

diff(a(t), t) = diff(a(t), t)

diff(b(t), t) = 0

diff(c(t), t) = -b(t)/c(t)

there is a diff(a(t), t) term 

how to plot this kind of system?

can diff(a(t), t) be ignore so that only consider two equations, diff(b(t),t) and diff(c(t),t) ?

if so, i use below to plot, it can not show the arrow clearing , i can only see arrow near origin, but not far point

with(plots):
fieldplot([0, y/x], x = -2 .. 2, y = -2 .. 2);
fieldplot([0, y/x], x = -2 .. 2, y = -2 .. 2, arrows = SLIM,grid = [1, 1]);
fieldplot([0, y/x], x = -10 .. 10, y = -10 .. 10);

 about how to calculate vector field of system of 3 differential equations which in terms of a(t), b(t), c(t) , diff(a(t),t), diff(b(t),t), diff(c(t),t)?

is the only method is the express diff(a(t),t), diff(b(t),t), diff(c(t),t) in terms of a(t), b(t), c(t) ?

for example 

<diff(a(t),t), diff(b(t),t), diff(c(t),t)>

if result is

diff(a(t),t) = a(t)*b(t)

diff(b(t),t) =b(t)*c(t)

diff(c(t),t) =c(t)*a(t)

<a(t)*b(t), b(t)*c(t), c(t)*a(t)>

then

is it the vector field <a*b, b*c, c*a> ?

sys := {diff(b(t),t) = 0,diff(c(t),t) = -b(t)/a(t)};
DEplot(sys, [b(t),c(t)], t=0..5, x=-5..5, y=-5..5);
Error, (in DEtools/DEplot) Option keyword (x) was not in the allowed set of options, consisting of: iterations, arrows, dirgrid, obsrange, scene, colour, linecolour, stepsize, a dependent variable range, a list of initial conditions or one of the allowed plot options: {animate, axes, color, colour, coords, font, scaling, style, symbol, title, view, animatecurves, animatefield, axesfont, dirfield, labelfont, linestyle, numframes, resolution, thickness, tickmarks, titlefont, xtickmarks, ytickmarks}, or one of the allowed dsolve/numeric options: {abserr, control, ctrl, initial, itask, maxder, maxfun, maxkop, maxord, maxpts, maxstep, method, mi..