The problem presented is a second-order linear differential equation:

$x^2 y'' - 3x y' + 4y = \ln(x) \quad \text{for} \quad x > 0$

This is a non-homogeneous linear differential equation. The general approach to solving such an equation involves two steps:

1. Solve the homogeneous equation $x^2 y'' - 3x y' + 4y = 0$ to find the complementary solution $y_c(x)$.
2. Find a particular solution $y_p(x)$ to the non-homogeneous equation $x^2 y'' - 3x y' + 4y = \ln(x)$.

Let's proceed step by step:

1. Solve the homogeneous equation:

The homogeneous equation is: $x^2 y'' - 3x y' + 4y = 0$

This is a Cauchy-Euler equation. To solve it, assume a solution of the form: $y(x) = x^r$

Substitute $y(x) = x^r$ into the homogeneous equation: $y' = r x^{r-1}, \quad y'' = r(r-1) x^{r-2}$

Substituting into the homogeneous equation: $x^2 \cdot r(r-1) x^{r-2} - 3x \cdot r x^{r-1} + 4x^r = 0$ $r(r-1) x^r - 3r x^r + 4x^r = 0$ $r(r-1) - 3r + 4 = 0$ Simplifying: $r^2 - r - 3r + 4 = 0$ $r^2 - 4r + 4 = 0$ $(r-2)^2 = 0$ Thus, $r = 2$ is a double root.

The general solution for the homogeneous equation is then: $y_c(x) = (C_1 + C_2 \ln(x)) x^2$ where $C_1$ and $C_2$ are constants.

2. Find a particular solution:

To find a particular solution to the non-homogeneous equation: $x^2 y'' - 3x y' + 4y = \ln(x)$

We can use the method of variation of parameters or undetermined coefficients. Since the right-hand side is $\ln(x)$, we propose a particular solution of the form: $y_p(x) = A \ln(x) + B$

Substitute $y_p(x)$ into the non-homogeneous equation to determine $A$ and $B$. I will now perform this step.

Let me calculate this.The general solution to the differential equation:

$x^2 y'' - 3x y' + 4y = \ln(x)$

is:

$y(x) = C_1 x^2 + C_2 x^2 \ln(x) + \frac{1}{4} \ln(x) + \frac{1}{4}$

where $C_1$ and $C_2$ are arbitrary constants determined by initial conditions (if given).

This solution consists of the complementary (homogeneous) part: $y_c(x) = C_1 x^2 + C_2 x^2 \ln(x)$ and the particular solution: $y_p(x) = \frac{1}{4} \ln(x) + \frac{1}{4}$

Would you like further details on any specific part, or do you have any other questions?

Here are 5 related questions you might want to explore:

1. How can initial conditions be used to determine $C_1$ and $C_2$?
2. What method would you use to solve this differential equation if the right-hand side was $x^3$ instead of $\ln(x)$?
3. Can you explain how variation of parameters works in solving non-homogeneous differential equations?
4. How does the method of undetermined coefficients differ from variation of parameters?
5. What are the specific applications of Cauchy-Euler equations in physics and engineering?

Tip: Always check the form of the non-homogeneous term to choose the appropriate method for finding a particular solution (e.g., undetermined coefficients or variation of parameters).