If jaconian matrix is zero, I guess you can't use this formula to get to the answer because you aren't allowed to put zero in Denominator.

Jacobian is akin to a Wronskian matrix and relates linear dependence of differential systems, in the case of a Jacobian, the relation or mapping is between two domains or representations of the system. If you get a zero, at points in time, then the system is exhibiting linear dependence. This could be due to a lack of adequate representation of the system, that is to say, you need to expand the representation of the system with more differentials, equations, whatever; but, could also indicate a nonlinear relationship, where parameters are becoming intertwined and dependent upon each other, lastly, it could also indicate an indeterminate, in other words, the system is breaking or becoming otherwise not adequately represented by your description, like in the case of an oscillating beam, if the beam breaks, then the equations fail to represent the system.

I had to solve a similar problem for nonlinear chemical equilibria. The standard approach only works for ideal behavior, because the free energy matrix becomes increasingly singular as the behavior departs from ideality. To get around this problem, you can solve a problem that is similar to the one you have started with. Add a little something to both sides of the equation before you try to solve it. In this case add a rank-one matrix to the Jacobian. Not just any rank-one matrix, but the one that maximizes the determinant of the resulting matrix, then you won't be dividing by zero. After you get the solution, you subtract a little something to get you back to the solution you wanted in the first place. This is a lot like adding one Lagrange Multiplier to a system. You have one more equation to solve, so it looks like you're getting farther from where you want to be, but it gets around the problem of the singular Jacobian. In your case you can probably show that this is equivalent to solving the transient system with a little added damping or non-zero viscosity.

Here's a link to an article. There are quite a few on the web...

http://www.sciencedirect.com/science/article/pii/S0893965907000614