Let $b : \mathbb R^N \rightarrow \mathbb R$ be a $C^1(\mathbb R^N)$ known function. We are looking for a solution $ u : \mathbb R^N \rightarrow \mathbb R$ of the following PDE :
$$\partial_1 u(x) + u(x)b(x) = 0 \ \ \ (1)$$ where $\partial_1 $ is the partial derivative of u with respect to $x_1$.
And so, more generally, let now $ c : \mathbb R^N \rightarrow \mathbb R^N$ be a $C^1(\mathbb R^N)$ vector field, we want to solve : $$\nabla u(x) + u(x)c(x) = 0 \ \ \ (2) $$
But I guess (2) follows from (1).
Thank you in advance for your answers.
Observe that the components $(x_2,x_3,\dots x_n)$ can be considered parameters, since your equation does not involve differentiation with respect to them. You can call $f(t)=u(t,x_2,x_3,\dots x_n)$ and $g(t)=b(t,x_2,x_3,\dots x_n)$. Your equation reads $$ f'(t)+f(t)g(t)=0 $$ which is linear and has solution $$ f(t)=Ce^{-\int_0^tg(s)ds} $$ Therefore, your original equation has solution $$ u(x_1,x_2,\dots x_n)=Ce^{-\int_0^{x_1}b(s,x_2,x_3,\dots x_n)ds} $$