The given differential equation is:

$(x_1 - y x_2) \, dy + (y^2 + x y^2) \, dx = 0$

This is a first-order partial differential equation. Let's break it down step by step.

Step 1: Check for the exactness of the equation

We first rewrite it in the form:

$M(x, y) \, dx + N(x, y) \, dy = 0$

Where:

$M(x, y) = y^2 + x y^2 \quad \text{and} \quad N(x, y) = x_1 - y x_2$

Now we check if the equation is exact by verifying if:

$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

Step 2: Calculate the partial derivatives

Partial derivative of $M(x, y)$ with respect to $y$:

$M(x, y) = y^2 + x y^2$ $\frac{\partial M}{\partial y} = 2y + 2xy$

Partial derivative of $N(x, y)$ with respect to $x$:

$N(x, y) = x_1 - y x_2$ $\frac{\partial N}{\partial x} = 0$

Step 3: Conclusion

The equation is not exact, because:

$\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$

Step 4: Use an integrating factor

Since the equation is not exact, we need to find an integrating factor to make it exact. One possible approach is to look for an integrating factor dependent on $x$ or $y$, which could simplify the equation.

Would you like me to proceed with the search for an integrating factor or discuss any other method for solving this equation?