plotting differential equation

i want to plot a solution of a differential equation which is given in the pic, i.e ‘phi vs t’ graph. where rho_0=2/3 and alpha=2enter image description here

The conditions are as follows,

case 1: ω = −1/3 and m = 1

case 2: ω = 1/3 and m = 1/2

case 3: ω = 0 and m = 2/3

case 4: ω = −1/2 and m = 4/3