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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.

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

Explore detailed solutions for non-exact differential equations using integrating factors, covering 20 problems, including initial value problems. Learn about the criteria for exactness and methods for finding integrating factors, suitable for undergraduate students.

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