The document you uploaded is a mathematical problem set titled "Tarea 7: Ecuaciones diferenciales no exactas y factores integrantes". It contains a series of differential equations to be solved using integrating factors. Specifically:

1. For problems 1 to 15, you are asked to:

   • Show that the given differential equations are not exact.
   • Find an integrating factor.
   • Solve the differential equation.

2. For problems 16 to 20, the task involves solving initial value problems.

Here’s a breakdown of the first problem as an example:

Problem 1:
$x\,dy - y\,dx = x^2 e^x \,dx$

Step 1: Verify if it's exact

For a differential equation $M(x, y)dx + N(x, y)dy = 0$, the condition for exactness is: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

In this case, $M(x, y) = -y$ and $N(x, y) = x^2 e^x - x$. We compute the partial derivatives to check if the equation is exact.

Step 2: Find an integrating factor

If the equation is not exact, an integrating factor is required. This can involve techniques like:

• Multiplying the equation by a function of $x$, $y$, or a combination of both.
• Using known forms or special methods to simplify the equation into an exact form.

Step 3: Solve the equation

Once the equation becomes exact, integrate both sides to find the solution.

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Would you like me to solve this problem or a specific one from the list? I can also assist with others or provide detailed steps. Let me know!

Here are 5 related questions to explore further:

1. What are the methods for identifying integrating factors in differential equations?
2. How can you verify if a differential equation is exact or not?
3. What is the general approach to solving initial value problems (IVPs)?
4. How do non-exact differential equations relate to physical problems?
5. What is the role of partial derivatives in solving differential equations?

Tip: When verifying exactness, always check if simplifications (like factoring or combining terms) make the equation easier to analyze.