Question

how that every member of the family of functions y = (4 ln(x) + C)/x is a solution of the differential equation x2y' + xy = 4. (Simplify as much as possible.)

Answer 1

$y=\dfrac{4\ln x+C}x\implies y'=\dfrac{x(4\ln x+C)'-(4\ln x+C)(x)'}{x^2}=\dfrac{4-4\ln x-C}{x^2}$

Substitute this into the differential equation:

$x^2y'+xy=4-4\ln x-C+4\ln x+C=4$

so every solution of the given form $y$ is a solution.