It holds that $$ M(x, y) d x+N(x, y) d y=0 $$ is in exact form if and only if $$ \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} $$
$$ \left(5 x^{4}-4 x^{3} y^{3}\right) d x+\left(5-3 x^{4} y^{4}\right) d y=0 $$ is exact if and only if
$$ \begin{aligned} &\frac{\partial }{\partial y}\left(5 x^{4}-4 x^{3} y^{3}\right)=\frac{\partial }{\partial x}\left(5-3 x^{4} y^{4}\right)\\ &\Leftrightarrow -12 x^{3} y^{2}=-12 x^{3} y^{2} \end{aligned} $$
Ok so I can prove that it is on exact form.
However I am unsure about how to solve it or whether I can get it simplified further. Can I only show it on implicit form? Also feel free to comment on my use of the if and iff arrows. I am not sure whether my notation is completely correct. $$ \begin{aligned} &\left(5 x^{4}-4 x^{3} y^{3}\right) d x+\left(5-3 x^{4} y^{4}\right) d y=0\\ &\Leftrightarrow \left(5 x^{4}-4 x^{3} y^{3}\right) d x=-\left(5-3 x^{4} y^{2}\right) d y\\ &\Rightarrow \int\left(5 x^{4}-4 x^{3} y^{3}\right) d x=-\int\left(5-3 x^{4} y^{2}\right) d y\\ &\Rightarrow x^{5}-x^{4} y^{3}+c_{1}=-5 y+x^{4} y^{3}+c_{2}\\ &\Leftrightarrow 5 y-2\left(x^{4} y^{3}\right)=-x^{5}-c_{1}+c_{2}\\ &\Rightarrow y\left(-2 x^{4} y^{2}+5\right)=-x^{5}+c \end{aligned} $$
Also I am not completely sure why it is useful to determine whether a function is in exact form.
$$\left(5 x^{4}-4 x^{3} y^{3}\right) d x+\left(5-3 x^{4} y^{4}\right) d y=0$$ $$ \frac{\partial }{\partial y}\left(5 x^{4}-4 x^{3} y^{3}\right)=\frac{\partial }{\partial x}\left(5-3 x^{4} y^{4}\right)\\ \implies-12 x^{3} y^{2}\color {red}{\ne}-12 x^{3} \color{red}{y^{4}} $$ This is not exact.