Hardcore Differentials!

Calculus Level 4

If the general solution of the differential equation,

\([y+x\sqrt { xy } (x+y)]dx\quad +\quad [y\sqrt { xy } (x+y)-x]dy\quad =\quad 0\)

is if the form ,

\(\frac { { x }^{ r }+{ y }^{ s } }{ t } +u\tan ^{ -1 }{ { \sqrt { \frac { x }{ y } } } } +\quad C\quad =\quad 0\)

where C is a constant , \(t,u\neq 1\)

and \(r,s,t,u>0\) then find the value of r+s+t+u.

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