We have the differential equation:

$x(x-1)y'' - (2x-1)y' + 2y = x^2(2x-3)$

To solve this second-order linear non-homogeneous differential equation, we'll break it down into several steps.

Step 1: Find the Homogeneous Solution

The corresponding homogeneous equation is:

$x(x-1)y'' - (2x-1)y' + 2y = 0$

This equation is a linear second-order differential equation with variable coefficients. We can try to solve it by assuming a solution of the form $y_h(x) = x^r$.

Substituting $y_h(x) = x^r$, $y_h'(x) = rx^{r-1}$, and $y_h''(x) = r(r-1)x^{r-2}$ into the homogeneous equation:

$x(x-1)r(r-1)x^{r-2} - (2x-1)rx^{r-1} + 2x^r = 0$

Simplify this expression:

$r(r-1)x^r(x-1) - r(2x-1)x^{r-1} + 2x^r = 0$

Expand and simplify further:

$r(r-1)x^r - r(r-1)x^{r-1} - 2r x^r + r x^{r-1} + 2x^r = 0$

Combine like terms:

$x^{r-1}[r(r-1) - r(2x-1) + 2x] = 0$

Since $x^{r-1} \neq 0$, we can set the polynomial coefficient to zero and solve for $r$:

$r(r-1) - r(2x-1) + 2x = 0$

This simplifies to a quadratic equation in $r$. Solve for $r$ to find the general solution of the homogeneous equation.

Step 2: Find the Particular Solution

The non-homogeneous term is $x^2(2x-3)$. We can try a particular solution of the form:

$y_p(x) = Ax^3 + Bx^2$

Substitute $y_p(x)$, $y_p'(x)$, and $y_p''(x)$ into the original non-homogeneous equation and solve for $A$ and $B$.

Step 3: General Solution

The general solution of the differential equation is:

$y(x) = y_h(x) + y_p(x)$

Where $y_h(x)$ is the solution to the homogeneous equation and $y_p(x)$ is the particular solution.

Final Steps:

• Solve the quadratic for $r$ to get $y_h(x)$.
• Substitute into the equation to find $A$ and $B$.
• Combine these results to write the general solution.

Would you like me to continue with the full detailed solution, including solving the quadratic for $r$ and finding $A$ and $B$?

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Here are five related questions:

1. What is the method for solving second-order linear differential equations with variable coefficients?
2. How do we determine the particular solution for non-homogeneous differential equations?
3. Can we always assume a solution of the form $y = x^r$ for homogeneous equations with variable coefficients?
4. What are the conditions for the existence of a unique solution to a second-order differential equation?
5. How does the presence of a non-homogeneous term affect the solution of a differential equation?

Tip: For solving differential equations with variable coefficients, the method of undetermined coefficients is often useful for finding particular solutions.