# Solving numerically an equation with an integral

I am trying to solve the following equation numerically:

Equation to solve for $$y$$:

$$na(ay-y)^{n}\int_{ay}^{+\infty} \frac{(x-y)^{-n}}{x-ay}dx=b$$

with for example $$a=4$$, $$n=1.25$$ and $$b=1.6$$.

I have adapted an answer to a similar question that I found on StackExchange:

Pdf[x_, y_, a_, n_] = n (x - y)^(-1 - n) ((-1 + a) y)^n
eps[x_, y_, a_] = (x - y)*a/(x - y*a)
Func[x_, y_, a_, n_] = FullSimplify[eps[x, y, a]*Pdf[x, y, a, n]]
f[y_?NumericQ] :=
NIntegrate[Func[x, y, 4, 1.25], {x, 4*y, Infinity}];
FindRoot[f[y] == 1.6, {y, 0}, WorkingPrecision -> 10000]


But I am getting a list of more than 10 error messages and the solution I get does not seem to be correct. My guess (from looking at plots of the integral NIntegrate[Func[x, y, 4, 1.25], {x, 4*y, Infinity}] which also generate loads of error messages) is that the solution is very small, that's why I am trying to search around 0. But I have no idea how to find it. Can anyone help please?

The integral is calculated exactly

f= Integrate[Func[x, y, a, n], x]

(*Out[]= (a n (x - y)^(
1 - n) ((-1 + a) y)^n Hypergeometric2F1[1, 1 - n, 2 - n, (
x - y)/((-1 + a) y)])/((-1 + a) (-1 + n) y)*)


But the integral diverges at the lower limit x=4ywith n=125/100,a=4. Check

f/. {x -> 4 y, a -> 4, n -> 125/100}

(*Out[]= -\[Infinity]*)

• Thank you very much! One quick question regarding your answer: when I use your code for the limit evaluation in my Mathematica, I get two error messages (Power::infy: Infinite expression 1/0 encountered) and then Out[]=ComplexInfinity. Is that because I may be using a different version of Mathematica or am I forgetting to load some particular package...? Thanks again! – Lednacek Oct 15 at 12:28
• I have 12.0.0 for Microsoft Windows (64-bit) (April 6, 2019). What do you have? – Alex Trounev Oct 15 at 12:33
• 11.3.0 for Mac OS (32-bit). I guess that explains it. Thanks again! – Lednacek Oct 15 at 12:34
• @Lednacek You're welcome! – Alex Trounev Oct 15 at 12:36