we will do seperation of variables
group y's with dy and x with dx
integrate both sides, and add constant to one side
use initial condiiton (if applcable)
solve for y
so
dy/dx=10(√x)/(e^y)
times both sides by e^y times dx
e^y dx=10(√x) dx
integrate both sides
an antiderivitive of 10(√x) is 10 times the antiderivitive of [tex]x^ \frac{1}{2}[/tex] which is [tex] \frac{x^\frac{3}{2}}{ \frac{3}{2} } [/tex] or [tex] \frac{2x^\frac{3}{2}}{3} [/tex]
e^y=[tex] \frac{20x^\frac{3}{2}}{3} [/tex] +C
take the ln of both sides
[tex]y=ln(\frac{20x^\frac{3}{2}}{3}+C)[/tex]
