Answer
$$y = \frac{{4{x^{3/2}}}}{3} + \frac{4}{3}$$
Work Step by Step
$$\eqalign{
& \left( {\text{b}} \right) \cr
& \frac{{dy}}{{dx}} = 2\sqrt x ,{\text{ }}\left( {4,12} \right) \cr
& {\text{Separate the variables}} \cr
& dy = 2\sqrt x dx \cr
& {\text{Integrate}} \cr
& \int {dy} = \int {2{x^{1/2}}} dx \cr
& y = 2\left( {\frac{{{x^{3/2}}}}{{3/2}}} \right) + C \cr
& y = \frac{{4{x^{3/2}}}}{3} + C{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Use the initial condition }}\left( {4,12} \right){\text{ to find the particular solution}} \cr
& 12 = \frac{{4{{\left( 4 \right)}^{3/2}}}}{3} + C \cr
& C = \frac{4}{3} \cr
& {\text{Substituting }}C{\text{ into }}\left( {\bf{1}} \right) \cr
& y = \frac{{4{x^{3/2}}}}{3} + \frac{4}{3} \cr
& \cr
& \left( {\text{a}} \right){\text{Slope field for the differential equation}} \cr
& \left( {\text{c}} \right){\text{Graph the solution}} \cr} $$
