This isn't a homework question, just something I'm trying to learn.
Consider the differential equation $\dot{y}=f(y)$ with known invariants $g(y)=$ Const , and assume that $g^{\prime}(y)$ has full rank. Prove by differentiation of the constraints that, for initial values satisfying $g\left(y_{0}\right)=0$, the solution of the differential-algebraic equation (DAE) $$ \begin{aligned} \dot{y} &=f(y)+g^{\prime}(y)^{T} \mu \\ 0 &=g(y) \end{aligned} $$ also solves the differential equation $\dot{y}=f(y)$.
I can't for the life of me understand how to do this. Any help would be appreciated.
Background: I'm taking a differential equations class. For a project we need to choose to implement something. I'm interested in dimensionality reduction so I decided to implement the algorithm in this paper. This involves solving a differential equation on the manifold of unitary matrices. I was reading background about how to do that in this textbook which is where I got this question from.