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Solve the differential equation to homogenous differential equation, show the complete solution for this

The given differential equation is:

\[
(xy + y^3) \, dx - x^2 \, dy = 0
\]

This is a first-order differential equation. To solve this, we'll follow these steps:

### Step 1: Rewrite the equation in standard form
Rearrange the equation as follows:

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

Dividing both sides by \(x^2 \, dx\), we get:

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

Simplify the right-hand side:

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

This equation is in the form:

\[
\frac{dy}{dx} = \frac{y

(xy + y^3)dx - x^2dy = 0. Solve the differential equation as a homogeneous differential equation.

Learn how to solve the homogeneous differential equation (xy + y^3)dx - x^2dy = 0. This solution involves rearranging the equation and simplifying using calculus methods. Suitable for undergraduate-level calculus students.

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