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The image contains a question asking to identify which differential equations are linear. Here are the equations provided:

1. \( \frac{d^3y}{dx^3} + \frac{dy}{dx} + y = \sqrt{x + 1} \)
2. \( \frac{dy}{dx} + y \sqrt{x^2 + 4} = x^2 \)
3. \( \frac{dy}{dx} + y^3 = 5 \)
4. \( x \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 5^x \)

### Explanation of Linearity:
A differential equation is **linear** if it can be written in the form:

\[
a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \ldots + a_1(x) \frac{dy}{dx} + a_0(x)y = g(x),
\]

where:
- \( a_i(x) \) are functions of \( x \),
- \( y \) and its derivatives appear only to the first power (i.e., no powers or products of \( y \) or its derivatives).

### Analysis of Each Equation:

1. **Equation 7:** \( \frac{d^3y}{dx^3} + \frac{dy}{dx} + y = \sqrt{x + 1} \)
   - This equation is **linear** because all terms involving \( y \) are to the first power.

2. **Equation 8:** \( \frac{dy}{dx} + y \sqrt{x^2 + 4} = x^2 \)
   - This equation is **linear** because \( y \) is to the first power, and it is multiplied by a function of \( x \).

3. **Equation 9:** \( \frac{dy}{dx} + y^3 = 5 \)
   - This equation is **nonlinear** because of the \( y^3 \) term, which is a power of \( y \).

4. **Equation 10:** \( x \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 5^x \)
   - This equation is **linear** because all terms involving \( y \) and its derivatives are to the first power.

### Conclusion:
The linear differential equations are:
- Equation 7
- Equation 8
- Equation 10

Would you like more details or have any questions? Here are some related questions:

1. What are the key characteristics that define a linear differential equation?
2. How do we identify nonlinearity in differential equations?
3. What are common methods to solve linear differential equations?
4. How does the order of a differential equation affect its linearity?
5. What are some real-world applications of linear differential equations?

**Tip:** Always check the powers and products of the dependent variable and its derivatives to determine the linearity of an equation.

Selecciona las ecuaciones diferenciales que son lineales.

Learn how to identify linear differential equations with a step-by-step explanation. This problem covers the analysis of differential equations and their classification as linear or nonlinear.

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