vdots f x mathbf mathbf ' mathbf '' ldots mathbf n end pmatrix begin pmatrix vdots end pmatrix math system form math mathbf f left x mathbf mathbf ' right boldsymbol math sources also require jacobian matrix math frac partial mathbf f x mathbf u mathbf v partial mathbf v math non singular order call implicit ode system implicit ode system satisfying jacobian non singularity condition transformed explicit ode system sources implicit ode systems singular jacobian termed differential algebraic equations daes distinction merely one terminology daes fundamentally different characteristics generally involved solve nonsingular ode systems ref
Formal science Mathematics Pure mathematics Analysis Ordinary differential equations