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Let's break down the problem step by step. The given differential equation is:

\[
\frac{dy}{dx} = \frac{-2xy + 24x}{x^2 + 16}
\]

We are also given the initial condition \( y(4) = 7.5 \). Now, let’s address each part of the problem:

---

### a) Plantear el problema de PVI (Planteamiento del Problema de Valor Inicial):

A **Problema de Valor Inicial (PVI)** involves solving a differential equation with an initial condition.

The PVI here consists of the differential equation:

\[
\frac{dy}{dx} = \frac{-2xy + 24x}{x^2 + 16}
\]

with the initial condition \( y(4) = 7.5 \).

---

### b) El PVI planteado tiene una única solución:

For the existence and uniqueness of the solution, we check the **Lipschitz condition** in the region containing the point \( x = 4 \). The given differential equation is of the form \( \frac{dy}{dx} = f(x, y) \), where:

\[
f(x, y) = \frac{-2xy + 24x}{x^2 + 16}
\]

We need to check whether \( f(x, y) \) and \( \frac{\partial f}{\partial y} \) are continuous around the point \( x = 4 \). Since both \( f(x, y) \) and its partial derivative with respect to \( y \):

\[
\frac{\partial f}{\partial y} = \frac{-2x}{x^2 + 16}
\]

are continuous for all \( x \) and \( y \), the **Picard-Lindelöf theorem** guarantees that the PVI has a unique solution.

---

### c) Encontrar la solución exacta del PVI:

We now solve the differential equation with the initial condition. First, the equation is separable, but it’s a first-order linear equation. The standard method to solve this is using an **integrating factor**. The equation can be rewritten as:

\[
\frac{dy}{dx} + \frac{2x}{x^2 + 16}y = \frac{24x}{x^2 + 16}
\]

This is in the form \( \frac{dy}{dx} + P(x)y = Q(x) \), where \( P(x) = \frac{2x}{x^2 + 16} \) and \( Q(x) = \frac{24x}{x^2 + 16} \).

The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{2x}{x^2 + 16} \, dx}
\]

Now, we can solve the integral for \( \mu(x) \):

\[
\int \frac{2x}{x^2 + 16} \, dx = \ln(x^2 + 16)
\]

So, the integrating factor is:

\[
\mu(x) = e^{\ln(x^2 + 16)} = x^2 + 16
\]

Multiplying through the original differential equation by the integrating factor \( x^2 + 16 \), we get:

\[
(x^2 + 16)\frac{dy}{dx} + 2xy = 24x
\]

This simplifies to:

\[
\frac{d}{dx} \left( (x^2 + 16)y \right) = 24x
\]

Now, integrating both sides with respect to \( x \):

\[
(x^2 + 16)y = \int 24x \, dx = 12x^2 + C
\]

Thus, the general solution is:

\[
y = \frac{12x^2 + C}{x^2 + 16}
\]

To find \( C \), we use the initial condition \( y(4) = 7.5 \):

\[
7.5 = \frac{12(4^2) + C}{4^2 + 16} = \frac{12(16) + C}{16 + 16} = \frac{192 + C}{32}
\]

Multiplying both sides by 32:

\[
240 = 192 + C
\]

Thus, \( C = 48 \).

Therefore, the exact solution to the PVI is:

\[
y = \frac{12x^2 + 48}{x^2 + 16}
\]

---

Would you like any further clarification on these steps or additional details on the methods used? 

Here are five related questions you might want to explore:

1. How do we verify the uniqueness of the solution using the Picard-Lindelöf theorem?
2. What is the significance of the integrating factor in solving first-order linear differential equations?
3. How can we extend this method to handle nonlinear differential equations?
4. Why is continuity of both \( f(x, y) \) and \( \frac{\partial f}{\partial y} \) important for the uniqueness of solutions?
5. What is the geometrical interpretation of the solution to the PVI?

**Tip:** When solving linear differential equations, always start by identifying if an integrating factor is applicable.

El cambio en el precio 'y' según el cambio en la cantidad demandada 'x', de una cierta mercancía, está dado por: \( \frac{dy}{dx} = \frac{-2xy + 24x}{x^2 + 16} \), con la condición inicial \( y(4) = 7.5 \).

Learn how to solve the differential equation that models the relationship between price and demand, given by \( \frac{dy}{dx} = \frac{-2xy + 24x}{x^2 + 16} \), with the initial condition \( y(4) = 7.5 \). This problem covers differential equations, integrating factors, and the Picard-Lindelöf theorem for uniqueness.

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