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To determine which of the given ordinary differential equations (ODEs) are linear, let's analyze each equation. A linear ODE is one where the dependent variable \( y \) and its derivatives appear to the first power and are not multiplied by each other or any nonlinear functions of \( y \).

1. \( \dot{y} = \cos(t + y) + t \)

   - **Nonlinear**: This equation is nonlinear because of the term \( \cos(t + y) \). The argument of the cosine involves both \( t \) and \( y \), making it nonlinear.

2. \( t \dot{y} = (\cos t) \dot{y} + \sin t \)

   - **Linear**: Simplifying, we get \( t \dot{y} = \cos(t) \dot{y} + \sin(t) \), which can be rearranged to:
     \[
     (t - \cos t) \dot{y} = \sin t
     \]
     Here, \( y \) and its derivative \( \dot{y} \) appear linearly, with no powers or products of \( y \). Therefore, this equation is linear.

3. \( y^{(3)} = y \ddot{y} + 2 e^t \dot{y} \)

   - **Nonlinear**: This equation is nonlinear because it includes the product \( y \ddot{y} \), where \( y \) multiplies its second derivative.

4. \( t^2 \ddot{y} = (y - 1) \log t \)

   - **Nonlinear**: This equation is nonlinear due to the presence of \( y - 1 \) on the right-hand side, as the equation depends nonlinearly on \( y \).

### Conclusion:
- **Linear ODE**: \( t \dot{y} = (\cos t) \dot{y} + \sin t \)
- **Nonlinear ODEs**: 
  1. \( \dot{y} = \cos(t + y) + t \)
  2. \( y^{(3)} = y \ddot{y} + 2 e^t \dot{y} \)
  3. \( t^2 \ddot{y} = (y - 1) \log t \)

Would you like further clarification on any of these ODEs or the concept of linearity?

Here are some related questions:
1. How can you solve linear ODEs analytically?
2. What is the significance of linearity in differential equations?
3. Can you classify ODEs based on order and linearity?
4. What are common methods to solve nonlinear ODEs?
5. How does the superposition principle apply to linear ODEs?

**Tip**: When analyzing an ODE for linearity, focus on whether the dependent variable and its derivatives appear only to the first degree and are not multiplied together.

Which of the following ODEs are linear?
1. ẏ = cos(t + y) + t
2. ty = (cos t) ẏ + sin t
3. y^(3) = y ÿ + 2e^t ẏ
4. t^2 ÿ = (y - 1) log t

Learn how to determine whether an ordinary differential equation (ODE) is linear or nonlinear. This example includes four ODEs and walks through the reasoning behind their classification based on their structure. Suitable for students studying Calculus or Differential Equations at the undergraduate level.

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