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.