k = 10^12;(*1*10^b;*) a = 1/(4*b^3*(1 - Cosh[b]*Cos[b])); a1 = (Cos[b*x] - Cosh[b*x]); a2 = (Sin[b*x] - Sinh[b*x]); phi1 = (Cos[b] - Cosh[b])*(Sin[b*(1 - i)] - Sinh[b*(1 - i)]); phi2 = (Sin[b] - Sinh[b])*(Cos[b*(1 - i)] - Cosh[b*(1 - i)]); phi3 = (Sin[b] + Sinh[b])*(Sin[b*(1 - i)] - Sinh[b*(1 - i)]); phi4 = (Cos[b] - Cosh[b])*(Cos[b*(1 - i)] - Cosh[b*(1 - i)]); eq = a ((phi1 - phi2)*(a1 /. x -> i) + (phi3 + phi4)*(a2 /. x -> i)); eq1 = 1 + (k*eq); i0 = Values@NSolve[(eq1 /. i -> .1) && 0 < b < 20, b][[All, 1]] sol = NDSolveValue[{D[eq1 /. b -> b[i], i], b[.1] == i0},     b, {i, $  MachineEpsilon, .5}, PrecisionGoal -> 16]; Plot[sol[x], {x, 0, .5}] 

What I am trying here is find the first few initial guess of roots, and using this initial roots to find the first few roots for different values i of from 0.01 to 0.5 using NDSolve