How do I find the analytical solution to this differential equation and then plot H vs t?

Question

(v₁²)/2g * (1 - 1/B⁴) = - (H + L)

where

v₁ = - dH/dt and H(to) = Ho

Answer 1

Assuming g, L and B are not functions of t, let K = √(2gB⁴/(B⁴-1)). Then we can use separation of variables

dH/dt = ±√(K/(H+L))

∫ √(H+L) dH = ±∫Kdt

(2/3)(H+L)^3/2 = ±Kt + C

H(t) = ((3/2)( ±Kt+C))^2/3

To solve for C:

C = (2/3)(H(t)+L)^3/2 - (±Kt)

C = (2/3)(H₀+L)^3/2 - (±K)t₀

H(t) = ((3/2)( ±Kt + ((2/3)(H₀+L)^3/2 - (±K)t₀)))^2/3

To Plot the Curve, need to know the values for g, B and L