Solve the following equation with RSolve and non-autonomous series?

Questions
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1

In this example, RSolve is not giving the result I’d expect (the parenthesis!). Am I missing something?

In[452]:= 
RSolve[{p[t + 1] == ( x[t] p[t] ) / ( 1 + b p[t] ), p[0] == p0}, p[t], 
  t] /. t -> 1

Out[452]= {{p[1] -> p0 / ( 1 + ( b p0 ) / x[0] )}}

I am using Mathematica 11.3 on Linux.

2

Hi

You have 2 variables p[t] and x[t] then you need a 2 equations to solve.

Regards,MI

3

The answer given by RSolve seems wrong to me, unless x[0]=1:

Clear[p1, p, p0];
eq = {p[t] == (x[t - 1] p[t - 1])/(1 + b p[t - 1]), p[0] == p0};
sol = RSolveValue[eq, p, t];
p1[0] = p0;
p1[t_] := (x[t - 1] p1[t - 1])/(1 + b p1[t - 1]);
Simplify[Table[sol[t] == p1[t], {t, 0, 3}]]
4

It is difficult to respond in the absence of a clear indication as to what was the expected result.

5

I would expect the result to be like this:

p1[0] = p0;
p1[t_] := (x[t - 1] p1[t - 1])/(1 + b p1[t - 1]);
6

Thanks Gianluca, that’s what I mean. From the recursion

p[t+1] = ( x[t] p[t] ) / ( 1 + b p[t] )
p[0] = p0

I expect

p[1] = ( x[0] p0 ) / ( 1 + b p0 )

but using RSolve I find

p[1] = ( x[0] p0 ) / ( x[0] + b p0 )