#### Answer

$${y^2} = 3{x^2} + 2x + C$$

#### Work Step by Step

$$\eqalign{
& \frac{{dy}}{{dx}} = \frac{{3x + 1}}{y} \cr
& {\text{Separating variables leads to}} \cr
& ydy = \left( {3x + 1} \right)dx \cr
& {\text{To solve this equation}}{\text{, determine the antiderivative of each side}} \cr
& \int {ydy} = \int {\left( {3x + 1} \right)dx} \cr
& {\text{integrating by using the power rule }} \cr
& \frac{{{y^2}}}{2} = 3\left( {\frac{{{x^2}}}{2}} \right) + x + C \cr
& \frac{{{y^2}}}{2} = \frac{3}{2}{x^2} + x + C \cr
& {\text{multiplying by 2}} \cr
& {y^2} = 3{x^2} + 2x + C \cr} $$