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Re: Plotting a series of Roots

So here I have an algorithm that gives me how my roots change as a function of lambda or t1[lambda,1]:

s1[lambda_, t_] = 
  x[t] /. DSolve[{x'[t] == lambda, x[0] == 0}, x[t], t][[1]];

lambda t;

t1[lambda_, value_] = t /. Solve[s1[lambda, t] == value, t][[1]];


s1[lambda, t1[lambda, value]];


Plot[t1[lambda, 1], {lambda, .1, 1}, PlotRange -> Automatic]

What I would like to do is in a similar fashion solve the differential equation bellow and plot how a particular root of that equation changes with \[Alpha]. It is important to note here that I have to use NDSolve rather than DSolve for this type of equation.What I would like is analogous to the above code modified to be t1[\[Alpha],1]with the modified equations.

M = 10000;
g = 1;
\[Alpha] = 1;
A = 10;
a = .01;(*initial plot temp*)
b = 1.2; (*final plot temp*)
c = 0.00000001; (*initial temp*)
d = 0; (*initial \[Beta]*)
m = 1;
bb = (g/(2*\[Pi])^(3/2)*(m/T)^(3/2)*E^(-m/T));

s = NDSolve[{\[Beta]'[T] == (3*\[Alpha]^2*M*T^(5/2))/(
      Sqrt[2*g]*m^(9/2))*(bb^2 - \[Beta][T]^2), \[Beta][c] == 
     d}, \[Beta], {T, a, b}, Method -> "BDF"];

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