Answer
$y=-ln [c-\dfrac{2}{5} (x-2)^{5/2}-\dfrac{4}{3} (x-2)^{3/2}]$
Work Step by Step
In order to to solve the given differential equation, we will have to separate the variables and then integrate.
Here, we have
$y'=xe^y\sqrt {x-2}$ or, $\int e^{-y} dy=\int x\sqrt {x-2} dx$
Plug in $x=p-2 \implies dx=dp$
Then, we have $ e^{-y} +c=\int (p+2) \sqrt p du$
or, $ e^{-y} =c-\dfrac{2}{5} p^{5/2}-\dfrac{4}{3} p^{3/2}$
and $ \ln e^{-y} =\ln [c-\dfrac{2}{5} p^{5/2}-\dfrac{4}{3} p^{3/2}]$
Hence, $y=-ln [c-\dfrac{2}{5} (x-2)^{5/2}-\dfrac{4}{3} (x-2)^{3/2}]$
