Answer
The particular solution that satisfies the intial condition is $y=(28-x^{\frac{3}{2}} )^{\frac{2}{3}}$
Work Step by Step
Start by integrating the differential equation using separation of variables
$\sqrt x +\sqrt y y' = 0$
$\int \sqrt y dy = \int - \sqrt x dx$
$ \frac{2}{3} y^{\frac{3}{2}} = -\frac{2}{3} x^{\frac{3}{2}} +C$
Let $\frac{3}{2} C = C'$
$y^{\frac{3}{2}} = - x^{\frac{3}{2}} +C'$
Use the initial condition to solve for C'
$ (\sqrt 9)^3 = -(\sqrt 1)^3 +C'$
Simplify
$C'= 28$
Substitute the C' value back into the general solution
$ y^{\frac{3}{2} }= -x^{\frac{3}{2}} +28$
$y=(28-x^{\frac{3}{2}} )^{\frac{2}{3}}$
