To solve the differential equation \((x^2D^2 + 5Dx - 5)y = 24x\ln x\), where \(D = \frac{d}{dx}\), follow these steps:

### Step 1: Rewrite the Differential Operator
Rewrite the operator in terms of \(D\):

\[
(x^2D^2 + 5xD - 5)y = 24x\ln x
\]

### Step 2: Find the Homogeneous Solution
First, solve the homogeneous equation:

\[
(x^2D^2 + 5xD - 5)y_h = 0
\]

Assume a solution of the form \(y_h = x^r\). Substitute \(y_h = x^r\) into the homogeneous equation:

\[
(x^2D^2 + 5xD - 5)x^r = 0
\]

Calculate \(D(x^r) = rx^{r-1}\) and \(D^2(x^r) = r(r-1)x^{r-2}\):

\[
x^2r(r-1)x^{r-2} + 5xr x^{r-1} - 5x^r = 0
\]

Simplify the equation:

\[
x^r[r(r-1) + 5r - 5] = 0
\]

Solve the characteristic equation:

\[
r(r-1) + 5r - 5 = 0
\]

\[
r^2 + 4r - 5 = 0
\]

\[
(r + 5)(r - 1) = 0
\]

Thus, \(r = -5\) or \(r = 1\). Therefore, the general solution to the homogeneous equation is:

\[
y_h = C_1x^{-5} + C_2x
\]

### Step 3: Find the Particular Solution
For the particular solution \(y_p\), use the method of undetermined coefficients. Assume a solution of the form:

\[
y_p = Ax\ln x + Bx
\]

Substitute \(y_p\) into the non-homogeneous equation. First, calculate the derivatives:

\[
D(y_p) = A\ln x + A + B
\]

\[
D^2(y_p) = \frac{A}{x}
\]

Substitute \(y_p\), \(D(y_p)\), and \(D^2(y_p)\) back into the original equation:

\[
(x^2 \cdot \frac{A}{x} + 5x(A\ln x + A + B) - 5(Ax\ln x + Bx)) = 24x\ln x
\]

Simplify each term:

\[
(Ax + 5xA\ln x + 5xA + 5xB - 5Ax\ln x - 5Bx) = 24x\ln x
\]

Combine like terms:

\[
Ax + 5xA + 5xB - 5Bx = 24x\ln x
\]

Simplify further:

\[
0x + 5xA = 24x\ln x
\]

Compare coefficients:

\[
5A = 24 \implies A = \frac{24}{5}
\]

Thus, the particular solution is:

\[
y_p = \frac{24}{5}x\ln x
\]

### Step 4: Combine Solutions
Combine the homogeneous and particular solutions:

\[
y = y_h + y_p = C_1x^{-5} + C_2x + \frac{24}{5}x\ln x
\]

### Final Solution
The general solution to the differential equation is:

\[
y = C_1x^{-5} + C_2x + \frac{24}{5}x\ln x
\]

Would you like details on any specific part of this solution, or do you have any questions?

Here are some related questions you might find interesting:

1. How can we determine the particular solution for different types of non-homogeneous terms?
2. What is the method of variation of parameters, and when is it used?
3. How does the solution change if the differential operator is different?
4. What are the applications of second-order linear differential equations in real-world problems?
5. Can you provide an example where the method of undetermined coefficients is not suitable?
6. How do you solve differential equations using Laplace transforms?
7. What are some common pitfalls in solving differential equations?
8. How does the characteristic equation help in solving differential equations?

**Tip:** When solving differential equations, always verify the solution by substituting it back into the original equation to ensure it satisfies the equation.

Learn how to solve the differential equation (x^2D^2 + 5Dx - 5)y = 24xlnx step-by-step. This problem involves using the method of undetermined coefficients and solving homogeneous equations. Suitable for advanced undergraduate students and those studying differential equations.

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