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The image contains a series of differential equations and their solutions. I'll walk you through the provided equations and their respective solutions:

### Problem 10:
\[
e^x \sin(2x) dx + \cos(x(e^x - y)) dy = 0
\]
**Solution:**
\[
-2 \cos x + e^x + y e^x + y e^x = c
\]

### Problem 11:
\[
(e^x + 1) e^x dx + (e^x + 1) e^x dy = 0
\]
**Solution:**
\[
(e^x + 1)^2 + 2(e^x + 1) = c
\]

### Problem 12:
\[
(y - y^2) \frac{dy}{dx} = (y + 1)^2
\]
**Solution:**
\[
-y (y + 1) + \ln |y| + 1 = \frac{1}{2} \ln \left| \frac{x + 1}{x - 1} \right| + c
\]

### Problem 13:
\[
\frac{dy}{dx} = \frac{xy + 3x - y - 3}{xy - 2x + 4y - 8}
\]
**Solution:**
\[
y - 5 \ln |y + 3| = x - 5 \ln |x + 4| + c
\]

### Problem 14:
\[
dy = \sin(x) \cos(2y - \cos^2 y) dx
\]
**Solution:**
\[
-\cot y = \cos x + c
\]

### Problem 15:
\[
x \sqrt{1 - y^2} dx = dy
\]
**Solution:**
\[
y = \sin \left( \frac{x^2}{2} + c \right)
\]

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Would you like a detailed breakdown of any specific equation or step? Here are 5 relative questions you might explore next:

1. What is the general process for solving first-order separable differential equations?
2. How can you derive the integrating factor for non-exact differential equations?
3. What substitution techniques can simplify trigonometric integrals in differential equations?
4. What conditions guarantee that a differential equation has an explicit solution?
5. How does logarithmic integration apply in equations like Problem 12?

**Tip:** When solving differential equations, always check for separability, which simplifies the problem significantly!

Solve a series of differential equations.

Explore step-by-step solutions to differential equations involving trigonometric identities and exponential functions. This problem set covers key calculus concepts such as separation of variables and integration by substitution, suitable for undergraduate students.

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