Problem Statement
Find the general solution of the differential equation:
$x \frac{dy}{dx} + 2y = x^2 \log x, \quad (x > 0)$
Verified Solution & Marking Scheme
Standard Form & Integrating Factor
Divide by $x$: $\frac{dy}{dx} + \frac{2}{x}y = x \log x$. $\text{I.F.} = e^{\int \frac{2}{x} dx} = x^2$.
Integrate by Parts
$y x^2 = \int x^3 \log x \, dx = \frac{x^4 \log x}{4} - \frac{x^4}{16} + C$
Divide by x²
$y = \frac{x^2}{4}\left(\log x - \frac{1}{4}\right) + \frac{C}{x^2}$