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We are given the following differential equation:

\[
\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y^2} + 4 \frac{\partial^2 f}{\partial x^2}
\]

and we need to determine which of the following functions satisfies this equation:

a) \( f(x, y) = x^2 y + x y^2 - y^3 \)

b) \( f(x, y) = x^2 y + x y^2 + x y^3 \)

c) \( f(x, y) = x^2 y - x y^2 - y^3 \)

d) \( f(x, y) = x^2 y - x y^2 - x y^3 \)

### Step-by-Step Approach

1. **Calculate the second mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y} \)** for each function.
2. **Calculate \( \frac{\partial^2 f}{\partial y^2} \)** and \( \frac{\partial^2 f}{\partial x^2} \)** for each function.
3. **Check which function satisfies the given equation.**

Let me perform the necessary calculations for each option.None of the provided functions (a, b, c, or d) directly satisfy the given differential equation. This suggests either there is an issue with the problem setup, or the equation might need additional context or corrections.

Would you like further clarifications on any of the steps or deeper insights into these calculations?

Here are five related questions you could explore:
1. How do you calculate mixed partial derivatives?
2. What are some common methods to solve partial differential equations?
3. How can we test if a function is a solution to a given equation?
4. Could transformations simplify solving such differential equations?
5. What role do boundary conditions play in differential equations?

**Tip:** In problems involving partial derivatives, always verify calculations to ensure accuracy in each derivative step.

¿Cuál de las siguientes funciones satisface la ecuación diferencial: (∂²f/∂x∂y) = (∂²f/∂y²) + 4 (∂²f/∂x²)? Seleccione una: a) f(x, y) = x²y + xy² - y³, b) f(x, y) = x²y + xy² + xy³, c) f(x, y) = x²y - xy² - y³, d) f(x, y) = x²y - xy² - xy³.

This problem asks which of four functions satisfies a specific partial differential equation involving mixed second-order derivatives. A detailed solution approach includes calculating second partial derivatives and verifying the function that satisfies the equation.

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