DSolve[{K'[t]-α σ K[t]==H0 (E^(μ t)),K[0]==K0},K[t],t]

Plot[Table[σ((((E^(t α σ)) (H0-(E^(t (μ-α σ))) H0-K0 μ+K0 α σ))/(-μ+α σ)))/(N0 (E^(ρ t)))/.{σ->0.3,ρ->0.03,H0->0,K0->1,N0->1,μ->0.1},{α,0,1,0.05}],{t,0,3}]