Solve the following differential equation: x ^ 2 * (d ^ 2 * y)/(d * x ^ 2) + 2x * d/dx (y) - 12y = x ^ 3 * log(x)

To solve the given differential equation

\[ x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} - 12y = x^3 \log(x), \]

we can follow these steps:

1. **Identify the type of differential equation**: This is a second-order linear differential equation with non-constant coefficients.

2. **Solve the homogeneous equation**: First, we solve the corresponding homogeneous equation

\[ x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} - 12y = 0. \]

3.

Learn how to solve the differential equation x^2 * (d^2 * y)/(d * x^2) + 2x * d/dx (y) - 12y = x^3 * log(x) with a detailed step-by-step solution. This problem involves solving a second-order linear differential equation with non-constant coefficients, suitable for advanced students and professionals.

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