To solve the equation $xy^2 + 2 = (3 - x^2 y) \frac{dy}{dx}$, we first rearrange it:

$(3 - x^2 y) \frac{dy}{dx} = xy^2 + 2$

Next, we can separate the variables:

$\frac{dy}{dx} = \frac{xy^2 + 2}{3 - x^2 y}$

To solve this differential equation, we can use the method of separation of variables or consider using an integrating factor, but given its form, separation seems straightforward.

1. Rearrange to separate $y$ and $x$: $\frac{dy}{xy^2 + 2} = \frac{dx}{3 - x^2 y}$

2. Integrate both sides: $\int \frac{dy}{xy^2 + 2} = \int \frac{dx}{3 - x^2 y}$

These integrals may require specific techniques or substitutions based on the terms involved.

Steps for Integration

1. Left Side: Depending on $x$, you may need partial fraction decomposition or substitution.
2. Right Side: Similar integration techniques might apply, possibly involving $x^2 y$.

Once the integrals are evaluated, you would need to combine the results and solve for $y$ as a function of $x$, if possible.

Would you like detailed steps on how to perform the integrals? Or do you have specific values for $x$ or $y$ to consider?

Questions to Consider

1. What methods have you used previously to solve differential equations?
2. Are you familiar with techniques like substitution or partial fractions?
3. Do you want to explore the implications of this equation in a specific context?
4. Would you like examples of similar differential equations and their solutions?
5. Are there particular conditions (like initial values) you want to apply?

Tip

Always check if the equation can be simplified before integrating; sometimes a clever substitution can make the process easier!