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You've uploaded a set of six differential equations. Here’s a breakdown of each question so I can assist further:

1. **Solve** \( x^2 y \, dx - (x^3 + y^3) \, dy = 0 \).
2. **Find the general solution** of \( \frac{dy}{dx} + xy = x e^{-x^2} y^{-1} \).
3. **Solve** \( \frac{dy}{dx} + y \cot x = \cos^3 x (1 + y^2 \sin^2 x) \), subject to \( y\left(\frac{\pi}{2}\right) = 1 \).
4. **Solve the differential equation** \( p^3 - 4xyp + 8y^2 = 0 \), where \( p = \frac{dy}{dx} \).
5. **Solve** \( \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 3y = \cos x + x^2 \).
6. **Solve** \( (x^2 D^2 - 3xD + 5)y = x^2 \sin(\log x) \), where \( D \equiv \frac{d}{dx} \).

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Learn how to solve the first-order differential equation x^2y dx - (x^3 + y^3) dy = 0 using the method of separation of variables. This problem covers key calculus concepts and is suitable for undergraduate-level mathematics students.

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