This is a very trivial matter, however given I have the vector field
$\frac{\partial \vec{v}}{\partial t} = e^t \hat k$
Integrating 2 ways:
don't factor $\hat k$:
$\vec{v} = \int e^t \hat k dt$
I would assume doing this way would be
$\vec{v} = e^t \hat k + \vec{c}(x,y,z)$
But Factoring $\hat k$:
$\vec{v} = \hat k \int e^t dt$
$\vec{v} = \hat k ( e^t + c(x,y,z))$
$\vec{v} = e^t \hat k + c(x,y,z) \hat k$
Which method is correct? Or are they both correct, yet way 2 forces it to be a specific solution to this differential equation.
Doing it 2 ways makes the vector constant have different directions? Ie, way 2, forces it to have only a k component.