Question

Please help with this 1st order DE: y'=3x^2-(y/x); y(1)=5

Answer 1

Use the formula! The solution is uniquely determined because of the initial value:
\[ y \left( x \right) =1/4\,{\frac {3\,{x}^{4}+17}{x}} \]

Answer 2

y' + (y/x) = 3x^2
integrating factor = e ^(INT(1/x) dx = e ^lnx = x
yx = INT3x^2*x dx
yx = 3x^4 / 4 + C
x=1 , y = 5
5 = 3/4 + c
c = 17/4

therefore
y = (1/4) (3x^4 + 17) / x