Solve the differential equation:

To solve the given differential equation, we'll first check if it's exact. If it's not exact, we'll try to make it exact by multiplying by an integrating factor.

Given the equation:

\[ (2xe^{y}y^{4}+2xy^{3}y)dx+(x^{2}y^{4}e^{y}-x^{2}y^{2}-3x)dy=0 \]

Let's check if it's exact by verifying if the mixed partial derivatives are equal:

\[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \]

Where \( M \) and \( N \) are the coefficients of \( dx \) and \( dy \) respectively.

\[ M = 2xe^{y}y^{4}+2xy^{3}y \]
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