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The problem displayed in the image is an exact differential equation (EDO) of the form:

\[
(4xy + x^2) \, dx + (2x^2 + 15y^2) \, dy = 0
\]

The solution is given as:

\[
Ax^2y + x^3 + By^3 = C
\]

where \( C \) is a constant, and \( A \) and \( B \) are the values to be determined.

We are also given:

- \( A = 2 \)
- \( B = 5 \)

The solution has blanks for the values of \( A \) and \( B \), which are correctly filled as:

- \( \text{En blanco 1} = 2 \)
- \( \text{En blanco 2} = 5 \)

Thus, the values provided are correct.

Would you like more details on how to solve such an exact differential equation?

Here are five related questions for further exploration:
1. How can you verify if a differential equation is exact?
2. What is the method to solve an exact differential equation?
3. How do constants like \( A \) and \( B \) influence the general solution of an equation?
4. What is the role of the integrating factor in solving non-exact differential equations?
5. Can the general solution of an exact differential equation change if initial conditions are provided?

**Tip:** When solving exact differential equations, always check if the mixed partial derivatives of the given functions are equal to ensure the equation is exact.

La solución de la EDO (4xy + x^2)dx + (2x^2 + 15y^2)dy = 0 es Ax^2y + x^3 + By^3 = C, donde A=2 y B=5

Solve the exact differential equation (4xy + x^2)dx + (2x^2 + 15y^2)dy = 0. Learn how to determine the constants A=2 and B=5 in the solution Ax^2y + x^3 + By^3 = C. This problem involves exactness conditions and step-by-step solving techniques.

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