Torricellis tratt (Separabel ODE) 

restart 

`:=`(ode, `*`(Pi, `*`(`^`(`+`(r, `/`(`*`(`+`(R, `-`(r)), `*`(y(t))), `*`(h))), 2), `*`(diff(y(t), t)))) = `+`(`-`(`*`(Pi, `*`(`^`(r, 2), `*`(sqrt(`+`(`*`(2, `*`(g, `*`(y(t)))))))))))) 

`*`(Pi, `*`(`^`(`+`(r, `/`(`*`(`+`(R, `-`(r)), `*`(y(t))), `*`(h))), 2), `*`(diff(y(t), t)))) = `+`(`-`(`*`(Pi, `*`(`^`(r, 2), `*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(g, `*`(y(t))), `/`(1, 2)))))))) (2.1)
 

`:=`(sol, dsolve(ode, y(t))) 

`+`(t, `/`(`*`(`+`(`*`(`^`(r, 2), `*`(`^`(h, 2))), `*`(`/`(2, 3), `*`(r, `*`(h, `*`(y(t), `*`(R))))), `-`(`*`(`/`(2, 3), `*`(`^`(r, 2), `*`(h, `*`(y(t)))))), `*`(`/`(1, 5), `*`(`^`(y(t), 2), `*`(`^`(R...
`+`(t, `/`(`*`(`+`(`*`(`^`(r, 2), `*`(`^`(h, 2))), `*`(`/`(2, 3), `*`(r, `*`(h, `*`(y(t), `*`(R))))), `-`(`*`(`/`(2, 3), `*`(`^`(r, 2), `*`(h, `*`(y(t)))))), `*`(`/`(1, 5), `*`(`^`(y(t), 2), `*`(`^`(R...
(2.2)
 

`:=`(eq1, eval(sol, {t = 0, y(t) = h})) 

`+`(`/`(`*`(`+`(`*`(`/`(8, 15), `*`(`^`(r, 2), `*`(`^`(h, 2)))), `*`(`/`(4, 15), `*`(r, `*`(`^`(h, 2), `*`(R)))), `*`(`/`(1, 5), `*`(`^`(h, 2), `*`(`^`(R, 2))))), `*`(`^`(2, `/`(1, 2)))), `*`(h, `*`(`... (2.3)
 

`:=`(_C1, solve(eq1, _C1)) 

`+`(`-`(`/`(`*`(`/`(1, 15), `*`(h, `*`(`+`(`*`(8, `*`(`^`(r, 2))), `*`(4, `*`(R, `*`(r))), `*`(3, `*`(`^`(R, 2)))), `*`(`^`(2, `/`(1, 2)))))), `*`(`^`(`*`(g, `*`(h)), `/`(1, 2)), `*`(`^`(r, 2)))))) (2.4)
 

sol 

`+`(t, `/`(`*`(`+`(`*`(`^`(r, 2), `*`(`^`(h, 2))), `*`(`/`(2, 3), `*`(r, `*`(h, `*`(y(t), `*`(R))))), `-`(`*`(`/`(2, 3), `*`(`^`(r, 2), `*`(h, `*`(y(t)))))), `*`(`/`(1, 5), `*`(`^`(y(t), 2), `*`(`^`(R...
`+`(t, `/`(`*`(`+`(`*`(`^`(r, 2), `*`(`^`(h, 2))), `*`(`/`(2, 3), `*`(r, `*`(h, `*`(y(t), `*`(R))))), `-`(`*`(`/`(2, 3), `*`(`^`(r, 2), `*`(h, `*`(y(t)))))), `*`(`/`(1, 5), `*`(`^`(y(t), 2), `*`(`^`(R...
(2.5)
 

`:=`(eq2, eval(sol, {t = T[epsilon], y(t) = epsilon})) 

`+`(T[epsilon], `/`(`*`(`+`(`*`(`^`(r, 2), `*`(`^`(h, 2))), `*`(`/`(2, 3), `*`(r, `*`(h, `*`(epsilon, `*`(R))))), `-`(`*`(`/`(2, 3), `*`(`^`(r, 2), `*`(h, `*`(epsilon))))), `*`(`/`(1, 5), `*`(`^`(epsi...
`+`(T[epsilon], `/`(`*`(`+`(`*`(`^`(r, 2), `*`(`^`(h, 2))), `*`(`/`(2, 3), `*`(r, `*`(h, `*`(epsilon, `*`(R))))), `-`(`*`(`/`(2, 3), `*`(`^`(r, 2), `*`(h, `*`(epsilon))))), `*`(`/`(1, 5), `*`(`^`(epsi...
(2.6)
 

`:=`(T[epsilon], solve(eq2, T[epsilon])) 

`+`(`-`(`/`(`*`(`/`(1, 15), `*`(`^`(2, `/`(1, 2)), `*`(`+`(`*`(15, `*`(epsilon, `*`(`^`(`*`(g, `*`(h)), `/`(1, 2)), `*`(`^`(r, 2), `*`(`^`(h, 2)))))), `*`(10, `*`(`^`(epsilon, 2), `*`(`^`(`*`(g, `*`(h...
`+`(`-`(`/`(`*`(`/`(1, 15), `*`(`^`(2, `/`(1, 2)), `*`(`+`(`*`(15, `*`(epsilon, `*`(`^`(`*`(g, `*`(h)), `/`(1, 2)), `*`(`^`(r, 2), `*`(`^`(h, 2)))))), `*`(10, `*`(`^`(epsilon, 2), `*`(`^`(`*`(g, `*`(h...
(2.7)
 

`:=`(T, limit(T[epsilon], epsilon = 0)) 

`+`(`/`(`*`(`/`(1, 15), `*`(`+`(`*`(8, `*`(h, `*`(`^`(2, `/`(1, 2)), `*`(`^`(r, 2))))), `*`(4, `*`(h, `*`(`^`(2, `/`(1, 2)), `*`(R, `*`(r))))), `*`(3, `*`(h, `*`(`^`(2, `/`(1, 2)), `*`(`^`(R, 2)))))))... (2.8)
 

eval(T, {R = `+`(`*`(15, `*`(r))), g = 9.81, h = .12}) 

`+`(`*`(5.478399186, `*`(`^`(2, `/`(1, 2))))) (2.9)
 

evalf(%, 3) 

7.73 (2.10)