Which of the following is an example of a second-order linear differential equation? Your written work for this problem will just be a short explanation.

		
(y')2 - 2y = 4

 		
4y'' - 3y' + 2y = 0

 		
fraction numerator d squared y over denominator d x squared end fraction equals x squared sin y

 		
y'' - y = 3yy'

A second-order linear differential equation has the general form:

\[
a(x) y'' + b(x) y' + c(x) y = f(x)
\]

where \( y \) is the dependent variable, \( x \) is the independent variable, and \( a(x) \), \( b(x) \), \( c(x) \), and \( f(x) \) are functions of \( x \) (they can be constants as well). The equation must be linear, meaning that \( y \), \( y' \), and \( y'' \) appear only to the first power and are not multiplied by each other.

### Analysis of the Options

1. **\((y')^2 - 2y = 4\)**  
   This is **not** linear because \( (y')^2 \) includes a square of the derivative.

2. **\(4y'' - 3y' + 2y = 0\)**  
   This is a **second-order linear differential equation** because it involves \( y'' \), \( y' \), and \( y \) to the first power, without any products or nonlinear terms.

3. **\(\frac{d^2 y}{dx^2} = x^2 \sin(y)\)**  
   This is **not** linear because \( y \) appears inside the \( \sin \) function, making it nonlinear.

4. **\(y'' - y = 3yy'\)**  
   This is **not** linear because of the term \( 3yy' \), which includes a product of \( y \) and \( y' \).

### Conclusion

The correct answer is:
**\(4y'' - 3y' + 2y = 0\)**  
This is the only option that is a second-order linear differential equation.

Would you like further details on differential equations?

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1. What makes a differential equation linear or nonlinear?
2. Can a higher-order differential equation also be linear?
3. How are second-order differential equations applied in physics?
4. What are some techniques for solving second-order linear differential equations?
5. Can a second-order linear differential equation have variable coefficients?

**Tip:** In a linear differential equation, the unknown function and its derivatives appear to the first power and are not multiplied by each other.

Which of the following is an example of a second-order linear differential equation?

Understand how to identify a second-order linear differential equation from given options. Learn about the form of a second-order linear equation and the conditions that define linearity. This problem explores key concepts in differential equations, suitable for students studying Calculus II or Differential Equations.

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