The image contains a series of differential equations. Let's go through each of them step by step.

1. Equation 1: $y' = \frac{3y}{x}$ This is a separable differential equation. We can solve it by separating the variables.

2. Equation 2: $y' = \frac{\cos 2x}{\sin 5y}$ This is another separable differential equation that can be solved by separating the $x$ and $y$ terms.

3. Equation 3: $y' = y \sin 3x$ This is a first-order linear differential equation, and it can be solved using an integrating factor or separation of variables.

4. Equation 4: $y' = \frac{e^{\frac{3y}{x}}}{x}$ This seems more complex due to the exponential term. We may need to simplify or apply a substitution method here.

5. Equation 5: $x^{-3} y' = e^{-4y}$ This can be simplified and then solved using separation of variables.

6. Equation 6: $y' = e^{x - 2y}, \quad y(0) = 0$ This is an initial value problem, and the equation can be solved using separation of variables or another technique.

7. Equation 7: $y' - 3x = xy$ This can be rewritten to isolate $y'$ on one side and then apply methods like separation or substitution.

8. Equation 8: $y dx = (x + 3) dy$ This is a form of an exact differential equation. We can check for exactness or solve it directly by integrating both sides.

Would you like a detailed step-by-step solution for any of these equations? If so, which one?

Here are 5 related questions you might want to explore:

1. How do we identify if a differential equation is separable?
2. What are the steps to solve an initial value problem (IVP)?
3. How can substitution techniques simplify solving non-linear equations?
4. What is the integrating factor method for first-order linear differential equations?
5. How do we check if a differential equation is exact?

Tip: Always check for symmetry or patterns in the equations to simplify your approach, especially with exponential and trigonometric terms.