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The image contains a set of differential equations with the instruction "Resolver las sigs. ecs. diferenciales" (Solve the following differential equations). There are 8 different problems provided. Here's a summary of each one:

1. \((e^x + 1) e^x \, dx + (x + 1) e^x \, dy = 0\)
   
2. \( y \, dy = 4x \, (y + 1) \, dx \)  
   With condition \( y(0) = 1 \)
   
3. \( (x - y) \, dx + x \, dy = 0 \)

4. \( y^2 \, dx + (x^2 + xy + y) \, dy = 0 \)  
   With condition \( y(0) = 1 \)

5. \( \left( \frac{1}{1 + \ln(x)} \right) dx = \frac{x}{(1 - \ln(x))} \, dy \)

6. \( (4y + 2x - 5) \, dx + (6y + 4x - 1) \, dy = 0 \)  
   With condition \( y(1) = 2 \)

7. \( (\cos(x)) \, dy + y \sin(x) = 1 \)

8. \( y'' = 2y + x(e^x + e^{2x}) \)  
   With condition \( y(0) = 2 \)

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This page provides step-by-step solutions to a variety of first and second-order differential equations, including initial value problems. Learn how to apply integration techniques and use key formulas to solve problems like y'' = 2y + x(e^x + e^{2x}) with conditions such as y(0) = 2.

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