https://file.aiquickdraw.com/mathbot/file/1725849148678sipkoltt.jpg

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.

The user submitted an image of multiple differential equations, including:
1. y' = 3y / x
2. y' = cos(2x) / sin(5y)
3. y' = y sin(3x)
4. y' = e^(3y / x) / x
5. x^(-3) y' = e^(-4y)
6. y' = e^(x - 2y), y(0) = 0
7. y' - 3x = xy
8. y dx = (x + 3) dy

This page explores solving various differential equations, including separable, first-order linear, and exact equations. We delve into topics like integrating factors, exponential functions, and solving initial value problems.

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