I've a question that asks:
Find the most general solution $Y^t = (y_1, y_2)$ to the linear system:
$\displaystyle \frac{dy^1}{dx}+3\frac{dy^2}{dx} = -4y^1-6y^2, ~ 2\frac{dy^1}{dx}+\frac{dy^2}{dx} = 7y^1+8y^2$
What exactly does the power on $y$ mean? Should $y^1$ should be $y_1$ and $y^2$ should be $y_2$?
In some cases, authors in analysis (like Fleming) choose to index the coordinates of vectors with upper indices, as in $$\mathbf x = (x^1,\dots,x^n)\in\mathbb R^n.$$ This is because it becomes useful in differential geometry to distinguish between upper and lower indices, and it just so happens that scalars that are the components of a vector require the upper indices. One must always be careful, in this setting, not to interpret upper indices as exponentiation.
I think what happened there is the exercise came from a book, or set of notes, that adopted this convention, and the author forgot to change it back to his own practice.