The question in the image asks to determine the solution of the given second-order differential equations (PD Orde 2) using the most appropriate method. It provides four options for equations and the definitions for the terms $Y' = \frac{dy}{dx}$ and $Y'' = \frac{d^2y}{dx^2}$.

Here is the breakdown of the options:

1. $Y'' + 3Y' + 2y = 5x^2 - 4x$
2. $XY'' + Y' + 3\cdot \frac{x^2}{x} = 2x$
3. $2XY'' + 2Y' - 2x = 3x^2$
4. $2Y'' + 3Y' - 2y = 4x - 9$

To determine the correct method, you'd typically use one of several techniques for solving second-order differential equations:

1. Homogeneous Equations (if the equation equals 0).
2. Non-Homogeneous Equations (in case of external functions like polynomials on the right-hand side).
3. Method of Undetermined Coefficients for specific non-homogeneous forms like polynomials or exponentials.
4. Variation of Parameters for more complex non-homogeneous equations.
5. Laplace Transform in some cases for more complex setups.

Each equation will be analyzed based on its structure to find the most suitable method for solving.

Would you like a detailed step-by-step solution or analysis of any particular option?

Here are some relative questions:

1. How do you solve a second-order homogeneous differential equation?
2. What is the method of undetermined coefficients, and when is it applied?
3. Can you explain how to solve non-homogeneous second-order differential equations?
4. What are the conditions for using the variation of parameters method?
5. How would Laplace transforms be useful in solving these kinds of equations?

Tip: When identifying the method to solve a differential equation, always check whether the equation is homogeneous or non-homogeneous and the form of the non-homogeneous part (like polynomials, exponentials, or trigonometric functions).