https://file.aiquickdraw.com/mathbot/file/1729126284859xdgkvxes.jpg

The image you provided contains four differential equations, each labeled from 1 to 4, and four direction fields (labeled A, B, C, D). The task is likely to match each differential equation to the correct direction field.

Let’s go through the analysis to match them:

1. **Equation 1: \( y' = x + 2y \)**
   - This is a linear differential equation. The slope depends on both \(x\) and \(y\), with a stronger dependence on \(y\) due to the factor \(2y\). 
   - We should expect the slope field to show diagonal patterns since \(y\) has a direct influence, while the dependence on \(x\) adds variation.

2. **Equation 2: \( y' = \frac{y^3}{6} - y - \frac{x^3}{6} \)**
   - This is a nonlinear equation, with terms involving cubic powers of both \(x\) and \(y\). The slope changes drastically for larger \(x\) and \(y\), so we expect more variation in the field and perhaps strong non-linear behaviors.

3. **Equation 3: \( y' = 2x - 1 - y^2 \)**
   - This equation has a quadratic dependence on \(y\) and a linear dependence on \(x\). The slope field should show behavior that changes rapidly for larger values of \(y\) because of the \(y^2\) term.

4. **Equation 4: \( y' = y(4 - y) \)**
   - This is a separable equation with no dependence on \(x\). The function \(y(4 - y)\) implies that the slope is zero when \(y = 0\) or \(y = 4\), which are equilibrium solutions. The field should show horizontal lines at \(y = 0\) and \(y = 4\), where the slope is zero.

### Now, analyzing the direction fields:

- **Field A:** Shows a saddle-like structure, with the slope behavior indicating two equilibrium points and changes based on \(y\), which fits **Equation 4** because it has equilibria at \(y = 0\) and \(y = 4\).
  
- **Field B:** Displays mostly vertical lines, indicating that the slope is primarily influenced by \(y\), which fits **Equation 1**.

- **Field C:** Exhibits changes in slope that suggest nonlinearity and rapid variation in the field, matching **Equation 3** due to the \(y^2\) term.

- **Field D:** Shows more complex variation in both the \(x\) and \(y\) directions, consistent with **Equation 2**, which has cubic terms for both variables.

### Matching:

1. **Equation 1: \( y' = x + 2y \)** matches with **Field B**.
2. **Equation 2: \( y' = \frac{y^3}{6} - y - \frac{x^3}{6} \)** matches with **Field D**.
3. **Equation 3: \( y' = 2x - 1 - y^2 \)** matches with **Field C**.
4. **Equation 4: \( y' = y(4 - y) \)** matches with **Field A**.

Would you like a deeper explanation of any of these matches, or do you have any further questions?

### Relative Questions:
1. Can you explain why equilibria points appear in direction fields?
2. How does the cubic dependence in a differential equation affect the direction field?
3. What role does the separability of a differential equation play in its slope field?
4. Why do linear terms in \(x\) and \(y\) produce predictable patterns in direction fields?
5. How would the direction field change if a higher-order term, like \(x^4\), were added?

### Tip: 
When interpreting direction fields, pay close attention to regions where the slope becomes zero, as these often correspond to equilibrium points in the differential equation.

The user provided four differential equations and corresponding direction fields. The task is to match each equation with its respective direction field.

Learn how to match four given differential equations to their corresponding direction fields. This exercise involves analyzing slope fields and equilibrium points in differential equations. Suitable for university-level students studying calculus or differential equations.

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