DSolve[{a'[t]==-k1 a[t], b'[t]==k1 a[t] - k2 b[t], c'[t]== k2 b[t], a[0]==1, b[0]==0, c[0]==0},{a[t], b[t], c[t]}, t]

a[t_, k1_]:= Exp[-k1 t];

b[t_, k1_, k2_]:= -(((E^(-k1 t-k2 t)) (-(E^(k1 t))+(E^(k2 t))) k1)/(k1-k2))

c[t_, k1_, k2_]:= (((E^(-k1 t-k2 t)) (-(E^(k1 t)) k1+(E^(k1 t+k2 t)) k1+(E^(k2 t)) k2-(E^(k1 t+k2 t)) k2))/(k1-k2))

Plot[{a[t,0.01],b[t,0.01,2],c[t,0.01,2]}, {t, 0, 100}]

s = DSolve[{a'[t] == -k32 a[t] -k31 a[t], b'[t] == k32 a[t] - k21 b[t], c'[t] == k31 a[t] + k21 b[t], a[0] == 0.5, b[0] == 0.3, c[0] == 0.2}, {a[t], b[t], c[t]}, t]

a[t_, k32_, k31_] := Evaluate[s[[1,1,2]]]

b[t_, k32_, k31_, k21_] := Evaluate[s[[1,2,2]]]

c[t_, k32_, k31_, k21_] := Evaluate[s[[1,3,2]]]

Plot[{a[t, 0.1, 0.1], b[t, 0.1, 0.1, 0.1], c[t, 0.1, 0.1, 0.1]}, {t, 0, 100}]

Animate[Plot[{a[t, k32, k31], b[t, k32, k31, k21], c[t, k31, k32, k21]}, {t, 0, 50}],{k32, 0, 2}, {k31, 0, 2},{k21, 0.1, 2},AnimationRunning->False]