# Solving non-linear equations: no methods available error

Posted 10 months ago
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 To determine the FWHM of LSFs I tried to solve the following non-linear equations with Mathematica: Assuming[a > 0 && Element[{a, x}, Reals], Sol = Solve[Abs[x] *BesselK[1, Abs[x]/a] == a/2, x]] Assuming[a > 0 && 0 <= Eta <= 1 && Element[{a, Eta, Ln2}, Reals] , Sol = Solve[ Eta/(a*Pi)*1/(1 + (x/a)^2) + (1 - Eta)/a* Sqrt[Log[2]/Pi] E^(-Log[2]*(x/a)^2) == Eta/(2*a*Pi) + (1 - Eta)/(2*a)*Sqrt[Log[2]/Pi] , x]] Mathematica states in both cases: Solve::nsmet: This system cannot be solved with the methods available to Solve.Is there any other way to solve these non-linear equations?
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Posted 10 months ago
 You can solve numerically.Try: With[{a = 1}, NSolve[Abs[x]*BesselK[1, Abs[x]/a] == a/2 && -10 < x < 10, x, Reals]] f[n_, a_] := FindRoot[Abs[x]*BesselK[1, Abs[x]/a] == a/2, {x, n}]; {f[-1, 1], f[1, 1]} And  With[{a = 1/2, Eta = 3/10}, NSolve[Eta/(a*Pi)*1/(1 + (x/a)^2) + (1 - Eta)/a* Sqrt[Log[2]/Pi] E^(-Log[2]*(x/a)^2) == Eta/(2*a*Pi) + (1 - Eta)/(2*a)*Sqrt[Log[2]/Pi] && -10 < x < 10, x, Reals]] g[n_, a_, Eta_] := FindRoot[Eta/(a*Pi)*1/(1 + (x/a)^2) + (1 - Eta)/a* Sqrt[Log[2]/Pi] E^(-Log[2]*(x/a)^2) == Eta/(2*a*Pi) + (1 - Eta)/(2*a)*Sqrt[Log[2]/Pi], {x, n}]; {g[-1, 1/2, 3/10], g[1, 1/2, 3/10]} 
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