Solve the First Order linear differential equation

xy' + y = x^2 lnx

xy' + y = x²*ln(x)

d/dx[yx] = x²*ln(x)

yx = ∫x²*ln(x) dx

For the integral on the right, IBP:

u = ln(x)

dv = x² dx

v = x³/3

du = dx/x

yx = x³/3*ln(x) - x³/9 + C

y = x²/3*ln(x) - x²/9 + C/x