Answer
$y= \frac{2}{5} (x-6)^{\frac{5}{2}} + 4(x-6)^{\frac{3}{2} } +C$
Work Step by Step
Find a general solution of the differential equation.
$\frac{dy}{dx} = x\sqrt {x-6} $
$ \int dy = \int x\sqrt {x-6} dx$
Let $u=x-6$ and $x=u+6$ and $du=dx$
Substitute these values into the integrand
$y = \int (u+6)u^{\frac{1}{2}}du$
$y= \int (u^{\frac{3}{2}} + 6u^{\frac{1}{2}}) du$
$y= \frac{2}{5} u^{\frac{5}{2}} + 4u^{\frac{3}{2}} +C$
$y= \frac{2}{5} (x-6)^{\frac{5}{2}} + 4(x-6)^{\frac{3}{2}} +C$, Resubstitute for u
