# Non-linear differential equation of first order with initial conditions

I need help with the following differential equation. I do not know how I can calculate this.

$\alpha'(t) = \alpha_1^{3/2} - \alpha^{3/2}(t) \qquad \text{with} \qquad \alpha(0)=0$

If you do wish to solve this problem with Mathematica, the approach is as follows.

a[t] /. Flatten@DSolve[{a'[t] == a1^(3/2) - a[t]^(3/2)}, a[t], t]
(* InverseFunction[(2 Sqrt[3] ArcTan[(1 + (2 Sqrt[#1])/Sqrt[a1])/Sqrt[3]] +
2 Log[Sqrt[a1] - Sqrt[#1]] - Log[a1 + Sqrt[a1] Sqrt[#1] + #1])/(3 Sqrt[a1]) &]
[-t + C[1]] *)


Apply the boundary condition to obtain the constant of integration.

C[1] -> Simplify[%[[0, 1]][0]]
(* C[1] -> π/(3 Sqrt[3] Sqrt[a1]) *)


%%[[0, 1]][a[t]] == First[%%] /. %