Solve (1 + xy) y dx + (1 - xy) dy = 0.

Steps to Solve

1. Rewrite the Differential Equation

We have the equation $(1 + xy)y , dx + (1 - xy) , dy = 0$. We can rewrite it as:

$$(1 + xy)y , dx + (1 - xy) , dy = 0.$$

This can be expressed as:

$$ (1 + xy)y , dx = -(1 - xy) , dy. $$

2. Separate Variables

Next, isolate $dy$ and $dx$:

$$ \frac{dy}{(1 - xy)} = -\frac{(1 + xy) y , dx}{(1 + xy)y}. $$

This simplifies to:

$$ \frac{dy}{(1 - xy)} = -\frac{y , dx}{(1 + xy)}. $$

3. Integrate Both Sides

Now, integrate both sides:

$$ \int \frac{dy}{(1 - xy)} = -\int \frac{y , dx}{(1 + xy)}. $$

The left side can be complex, and for the right side, it's also integrated concerning $x$. We know that integrating both sides yields:

Let’s solve separately:

4. Solve the Integrals

For the left integral:

Let $u = 1 - xy \implies du = -y , dx$:

$$ \int \frac{dy}{(1 - xy)} = -\int \frac{dy}{u} = -\ln |u| + C = -\ln |1 - xy| + C.$$

For the right integral, solving yields an implicit function of $x$ and $y$:

$$ -\int y dx = -y x + C. $$

5. Combine Results and Solve for y

On combining results from both integrals, we would generally extract $y$ in terms of $x$.

Thus, we arrive at:

$$ -\ln |1 - xy| = -y x + C. $$

This represents the implicit solution which can be rearranged as:

$$ |1 - xy| = e^{-C} e^{-yx} $$

which expresses the solution to our differential equation.