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.