Specify all  linear equations from the list below.

Select one or more:

u′′=t2u″=t2

sinu′=usin⁡u′=u 

u′′−u′=0u″−u′=0 

u′=t2+2u′=t2+2

(u2)′+u=1


To identify the linear equations from the given list, let's analyze each one:

1. **\( u′′ = t^2 \)**
   - This is a **linear** equation because it can be written as \( u'' = t^2 \), where \( u'' \) is linear in \( u \).

2. **\( \sin(u′) = u \)**
   - This is **non-linear** because the sine function is applied to \( u' \).

3. **\( u′′ − u′ = 0 \)**
   - This is a **linear** equation because it is a linear combination of the derivatives \( u'' \) and \( u' \).

4. **\( u′ = t^2 + 2 \)**
   - This is a **linear** equation because it can be written as \( u' = t^2 + 2 \), and \( u' \) is linear in \( u \).

5. **\( (u^2)′ + u = 1 \)**
   - This is **non-linear** because the derivative is applied to \( u^2 \), making it non-linear in \( u \).

### Summary:
The linear equations from the list are:
- \( u′′ = t^2 \)
- \( u′′ − u′ = 0 \)
- \( u′ = t^2 + 2 \)

Would you like more details or have any questions? Here are 5 related questions to deepen your understanding:

1. How does the presence of trigonometric functions affect the linearity of differential equations?
2. What are the characteristics of linear differential equations that make them easier to solve?
3. Can a differential equation involving a non-linear term like \( u^2 \) ever be linear?
4. How would the classification change if the right-hand side of an equation was a function of the dependent variable?
5. What are some examples of real-world systems that can be modeled using linear differential equations?

**Tip:** Always check whether the dependent variable or its derivatives are raised to a power or involved in non-linear operations to determine linearity.

Learn how to identify linear differential equations from a list of examples. This problem covers differential equations, linearity, and trigonometric functions. Suitable for advanced mathematics students and those interested in differential equations.

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