Answer
$${y^2} - y = \frac{{{x^3}}}{3} + 5x + 110$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = \frac{{{x^2} + 5}}{{2y - 1}};\,\,\,\,\,\,\,\,\,\,\,y\left( 0 \right) = 11 \cr
& {\text{Separating variables leads to}} \cr
& \left( {2y - 1} \right)dy = \left( {{x^2} + 5} \right)dx \cr
& {\text{To solve this equation}}{\text{, determine the antiderivative of each side}} \cr
& \int {\left( {2y - 1} \right)dy} = \int {\left( {{x^2} + 5} \right)} dx \cr
& {\text{integrate by using the power rule}} \cr
& {y^2} - y = \frac{{{x^3}}}{3} + 5x + C\,\,\,\,\,\,\,\,\,\,\,\,\left( {\bf{1}} \right) \cr
& \cr
& {\text{we can find the constant }}C{\text{ using the initial value problem}}\,\,\,y\left( 0 \right) = 1 \cr
& \,y\left( 0 \right) = 11{\text{ implies that }}y = 11{\text{ when }}x = 0 \cr
& {\text{substituting these values into }}\left( {\bf{1}} \right) \cr
& {\left( {11} \right)^2} - 11 = \frac{{{{\left( 0 \right)}^3}}}{3} + 5\left( 0 \right) + C \cr
& C = 110 \cr
& {\text{substitute }}C = 110{\text{ into }}\left( {\bf{1}} \right){\text{ to find the particular solution}} \cr
& {y^2} - y = \frac{{{x^3}}}{3} + 5x + 110 \cr} $$
