As the question says, why are exact differential equations called so?
From Wikipedia, I got "The nomenclature of "exact differential equation" refers to the exact differential of a function". That leads me to ask why is an exact differential called so? Usually, the term 'exact' in the context of English refers to some quantity that is not approximated in any way. How does that definition fit here?
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$\begingroup$A differential $k$ - form $\omega$ is exact if there exists a $(k-1)$ - form $\alpha$ such that $\omega=d \alpha$. This (applied to your context) motivates the name 'exact'.
EDIT: to give an example, consider the equation:$$P(x,y) dx + Q(x,y) dy =0 $$This equation is exact if there is some $U$ such that:$$\frac{\partial U}{\partial x} = P$$and$$\frac{\partial U}{\partial y} = Q$$Notice that:$$ dU = \frac{\partial U}{\partial x} dx +\frac{\partial U}{\partial y} dy = P dx +Qdy$$That is, $U$ is an exact differential form.
$\endgroup$ 4 $\begingroup$An exact differential equation is defined as an equation which has a solution of the form:$$du(x,y)=P(x,y)dx+Q(x,y)dy$$ if the DE is defined as:$$P(x,y)dx+Q(x,y)dy=0$$ leading to the general solution of:$$u(x,y)=C$$It may be called an exact equation because it is based on the requirements of continuous pds or that the value of the constant can be worked out easily so values given are "exact" rather than how PDEs are often solved using numerical methods which are effectively good approximations
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