Determine the ODE (xy + 2y³) dx - x³dy = 0 and hence, solve the given equation

To determine and solve the given ordinary differential equation (ODE) \((xy + 2y^3) \, dx - x^3 \, dy = 0\), let's proceed step-by-step:

### 1. Express the ODE in Differential Form:
Given the ODE:  
\[
(xy + 2y^3) \, dx - x^3 \, dy = 0
\]

Rewrite it as:  
\[
(xy + 2y^3) \, dx = x^3 \, dy
\]

Or, equivalently:  
\[
\frac{dy}{dx} = \frac{xy + 2y^3}{x^3}
\]

### 2. Separate the Variables:
Separate the variables to make it easier to integrate. We can rewrite the equation as:  
\[
\frac{dy}{y(y^2 + \frac{x}{2})} = \frac{dx}{x^2}
\]

Or, breaking it into partial fractions:  
\[
\frac{1}{y^2} - \frac{1}{y^2 + \frac{x}{2}} \, dy = \frac{dx}{x^2}
\]

### 3. Integrate Both Sides:
Integrate each side separately.

#### Integrate the left side:
\[
\int \left( \frac{1}{y^2} - \frac{1}{y^2 + \frac{x}{2}} \right) \, dy
\]

The integrals are:  
\[
- \frac{1}{y} - \frac{1}{2} \ln |y^2 + \frac{x}{2}|
\]

#### Integrate the right side:
\[
\int \frac{1}{x^2} \, dx = -\frac{1}{x}
\]

### 4. Combine the Results:
Combining the results of the integrals:  
\[
- \frac{1}{y} - \frac{1}{2} \ln |y^2 + \frac{x}{2}| = -\frac{1}{x} + C
\]

Where \(C\) is the constant of integration. Let's multiply the entire equation by \(-1\) to simplify it:

\[
\frac{1}{y} + \frac{1}{2} \ln |y^2 + \frac{x}{2}| = \frac{1}{x} + C
\]

### 5. Solve for the General Solution:
Since solving this equation explicitly for \(y\) might be complex due to the logarithmic term, we can express the implicit solution:

\[
\frac{1}{y} + \frac{1}{2} \ln |y^2 + \frac{x}{2}| - \frac{1}{x} = C
\]

This is the implicit solution to the given ODE.

### Summary:
The given differential equation \((xy + 2y^3) \, dx - x^3 \, dy = 0\) can be solved to yield an implicit solution:

\[
\frac{1}{y} + \frac{1}{2} \ln |y^2 + \frac{x}{2}| = \frac{1}{x} + C
\]

### Follow-up Questions:
1. What is the significance of the constant \(C\) in the solution?
2. How can this implicit solution be used to find a particular solution given an initial condition?
3. What are the potential methods to solve this ODE explicitly for \(y\)?
4. How does this method of solving compare to other techniques like using an integrating factor?
5. Can this ODE be classified into a specific type, like separable, exact, or linear?

### Tip:
When solving ODEs, always check if you can simplify the equation by separating variables or finding an integrating factor, as it can make integration more straightforward.

Determine the ODE (xy + 2y³) dx - x³ dy = 0 and hence, solve the given equation

Explore the step-by-step solution of the ODE (xy + 2y³) dx - x³ dy = 0 using separation of variables and integration techniques. This guide covers the derivation of the implicit solution and partial fraction decomposition, suitable for advanced calculus students.

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