For given values of b, d, and L you have a standard system of linear equation. An obvious solution to these equations are all ci = 0. Each equation will be an equation for a plane in the three-dimensional space for c1, c2, c4. From there you can use geometric arguments that either 2 planes are parallel and doesn't intersect or intersects along a line, in the second case such a line either intersects the third plane, lies in the third plane or does not intersect the plane, the third case is out since we know that [0 0 0] is a solution, therefor either [0 0 0] is the only solution or lies on the line. What you'll have to do is to check whether there is a line of solutions or the one and only solution.