cmath11.nb - Ordinary differential equations 
DSolve and General/partial analytic solutions

ODE and General analytic Solution

Clear["Global`*"]

Definition of ODE (a system with grow and saturation)

deq1=x'[t]==k x[t](P-x[t])

General solution

DSolve[deq1,x[t],t]

DSolve[deq1,x,t]

Solution as user's function

sol=DSolve[deq1,x,t]
xt=x[t] /.sol[[1]]

Initial conditions and partial solutions

Clear["Global`*"]
k=1/100000; P=200000;
deq2=x'[t]==k x[t](P-x[t])

Initial value problem  x[t0]=x0,

x0=10000;
IVP={deq2, x[0]==x0}

Partial Solution

sol=DSolve[IVP,x,t]
xt1= x[t]/.sol[[1]]

Plot of solution

graph1=Plot[xt1,{t,0,5}, PlotStyle->Red]

Another Partial Solution

sol=DSolve[{deq2,x[0]==50000},x,t];
xt2=x[t]/.sol[[1]]
graph2=Plot[xt2,{t,0,5}, PlotStyle->Blue]

Show[{graph1,graph2}]

General solution of an initial value problem

Partial solution  (x[t0]=x0)

Clear[x0]
sol=DSolve[{deq2,x[0]==x0},x,t]

Some solutions   (x[0]=x0, x'[0]=v0)

xt=x[t]/.sol[[1]]

xt1=xt /.{x0->10000}
xt2=xt /.{x0->20000}
xt3=xt /.{x0->30000}
xt4=xt /.{x0->40000}
tmin=-2;
tmax=5;
Plot[{xt1,xt2,xt3,xt4},{t,tmin,tmax}]