DSolve[{x'[t]+x[t]==0,x[0]==1},x[t],t]

DSolve[{x'[t]+3 x[t]==Exp[-4 t],x[0]==3},x[t],t]

DSolve[{x'[t]+t x[t]==Exp[-t]},x[t],t]

DSolve[{x''[t]+x[t]==1,x[0]==1,x'[0]==5},x[t],t]

Plot[x[t],{t,0,3}]

losung=x[t]/.DSolve[{x''[t]+x[t]==1,x[0]==1,x'[0]==2},x[t],t]

{{x[t]->1+2 Sin[t]}}
Plot[losung,{t,0,3},PlotRange->{0,5}]

losung=x[t]/.DSolve[{x''[t]+x[t]==1,x[0]==1,x[3]==2},x[t],t]

{1+2 Sec[3] Sin[t]}
Plot[losung,{t,0,3}, PlotRange->{0,9}]

llll=x[t ]/.DSolve[{x''[t]+2 x'[t]+3==Sin[t],x[0]==2,x'[0]==5},x[t],t]
Plot[llll,{t,0,10}]

Yule Process

i=1

b=1

p1[t_]=Exp[-b t]

p2[t_]=b Exp[-2 b t] Integrate[Exp[2 b r]p1[r],{r,0,t}]

p3[t_]=b Exp[-3 b t] Integrate[Exp[3 b r]p2[r],{r,0,t}]

p4[t_]=b Exp[-4 b t] Integrate[Exp[4 b r]p3[r],{r,0,t}]

Plot[{p1[t],p2[t],p3[t],p4[t]},{t,0,10},PlotRange->{0,1}]

Der reine Todesprozess

μ=1

Clear[g4]

DSolve[{g4'[t]==-μ g4[t],g4[0]==1},g4[t],t]

g4[t_]=(E^(-t μ))

Clear[g3]

DSolve[{g3'[t]==-μ g3[t]+μ g4[t],g3[0]==0},g3[t],t]

g3[t]=(E^(-t)) t

g3[t_]=(E^(-t μ))t

Clear[g2]

DSolve[{g2'[t]==-μ g2[t]+μ g3[t],g2[0]==0},g2[t],t]

g2[t_]=(1/2) (E^(-t )) (t^2)

Clear[g1]

DSolve[{g1'[t]==-μ g1[t]+μ g2[t],g1[0]==0},g1[t],t]

g1[t]=(1/6) (E^(-t )) (t^3)

Plot[{g1[t],g2[t],g3[t],g4[t]},{t,0,10}, PlotRange->{0,1}]

sum[t_]=g1[t]+g2[t]+g3[t]+g4[t]

(E^(-t))+(E^(-t)) t+(1/2) (E^(-t)) (t^2)+(1/6) (E^(-t)) (t^3)
g0[t_]=1-sum[t]

Plot[sum[t],{t,0,10}]

Plot[{g0[t],g1[t],g2[t],g3[t],g4[t]},{t,0,10}, PlotRange->{0,1}]