Solve the differential equation by suitable method. ydxxdy = xy² dy + y³dx

To solve the given differential equation:

\[
y \, dx \, dx \, dy = x y^2 \, dy + y^3 \, dx,
\]

we first rewrite the equation in a more standard differential form:

\[
y \, \frac{d^2x}{dy^2} \, \frac{dy}{dx} = x y^2 \, \frac{dy}{dx} + y^3 \, \frac{dx}{dy}.
\]

This equation looks quite complex and involves both \(x\), \(y\), and their differentials. Let's rewrite the equation in a more recognizable form that is easier to solve:

\[
y \, \frac{dx}{dy} \, dy = x y^2 \, dy + y^3 \, dx.
\]

Let's look at how to solve this equation systematically. We will use an appropriate method to handle this by separating variables where necessary. Notice that some terms can be factored out as products of functions of \(x\) and \(y\).

Now let's rewrite the equation:

\[
\text{Re-organize the terms in the equation. Next, Simplify differential, the required canonical state of the equation to fit into a respective category like simple polynomials, log expressions, etc}
- Try multiplying and dividing, applying the derivative formula for 1 to get the better simplified equation.

Explore how to solve the differential equation ydxxdy = xy² dy + y³dx step-by-step with a detailed solution. This problem involves advanced concepts in differential equations, including separation of variables and integration techniques. Suitable for advanced undergraduate students and beyond.

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