Problem 10 | Separation of Variables

$v (dv / dx) = g$

$v \, dv = g \, dx$

$\displaystyle \int v \, dv = g \int dx$

$\dfrac{v^2}{2} = gx + \dfrac{c}{2}$

$v^2 = 2gx + c$
 

when x = x_o, v = v_o
${v_o}^2 = 2gx_o + c$

$c = {v_o}^2 - 2gx_o$
 

thus,
$v^2 = 2gx + ({v_o}^2 - 2gx_o)$

$v^2 - {v_o}^2 = 2gx - 2gx_o$

$v^2 - {v_o}^2 = 2g(x - x_o)$           answer