In the standard expanding variety model where the representative household faces the following CRRA utility function
$$ \int_{0}^{\infty} \exp(-\rho t) \frac{C(t)^{1-\theta}-1}{1-\theta}dt $$
and final goods that is produced competitively with the following production $$ Y(t) = \frac{1}{1-\beta} \left( \int_{0}^{N(t)}x(v,t)^{1-\beta}dv\right)L^{\beta} $$
with the following resource constraint
$$ C(t) + X(t) + Z(t) \leq Y(t) $$
where $X(t)$ is the total spending on machines, and $Z(t)$ is the total expenditure on research and development.
The innovation possibilities frontier takes the form
$$ \dot{N}(t) = \eta Z(t) $$
The intermediary goods firm faces the following problem where
$$ r(t) V(v,t) - \dot{V}(v,t) = \pi(v,t) $$
The thing I don't understand about the model is how the Euler Equation is derived. In almost everywhere I look the Euler Equation is the standard
$$ \frac{\dot{C}(t)}{C(t)} = \frac{1}{\theta}(r(t)-\rho) $$
I don't understand how you would derive this equation.