## 16.1 Notation

A DAE is defined by a set of expressions for the residuals of differential equations and algebraic equations $$r(y', y, t, \theta)$$, and consistent initial conditions $$y(t_0, \theta) = y_0, y'(t_0, \theta)=y'_0$$. The DAE is define by residual function as $$r(y', y, t, \theta)=0$$. The $$\theta$$ dependence is included in the notation to highlight that the solution $$y(t)$$ is a function of any parameters used in the computation.