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Re: How to find the roots of non linear equations

• To: mathgroup at smc.vnet.net
• Subject: [mg120023] Re: How to find the roots of non linear equations
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Tue, 5 Mar 2013 04:19:45 -0500 (EST)
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• References: <20130304035756.9C46A67E5@smc.vnet.net>

Multiple equations need to be entered as a list

f1 = a*Sin[x]/l0 + b*Sin[x]/l1;
f2 = y*Cos[x]/l0 + y*Cos[x]/l1 - (1/e^y);
soln1 = Solve[{f1 == 0, f2 == 0}, {x, y}] // Simplify

Solve::ifun: Inverse functions are being used by Solve, so some
solutions may not be found; use Reduce for complete solution
information. >>

{{x -> -ArcCos[(-l0 - l1)/
             Sqrt[(l0 + l1)^2]],
     y -> ProductLog[
           -((l0*l1*Log[e])/
                Sqrt[(l0 + l1)^2])]/
         Log[e]},
   {x -> ArcCos[(-l0 - l1)/
           Sqrt[(l0 + l1)^2]],
     y -> ProductLog[
           -((l0*l1*Log[e])/
                Sqrt[(l0 + l1)^2])]/
         Log[e]},
   {x -> -ArcCos[(l0 + l1)/
             Sqrt[(l0 + l1)^2]],
     y -> ProductLog[
           (l0*l1*Log[e])/
             Sqrt[(l0 + l1)^2]]/
         Log[e]},
   {x -> ArcCos[(l0 + l1)/
           Sqrt[(l0 + l1)^2]],
     y -> ProductLog[
           (l0*l1*Log[e])/
             Sqrt[(l0 + l1)^2]]/
         Log[e]}}


Verifying

{f1 == 0, f2 == 0} /. soln1 // Simplify

{{True, True}, {True, True}, {True, True}, {True, True}}


Or if the denominators are intended to be the numbers 10 and 11 and e
is intended to be E

f1 = a*Sin[x]/10 + b*Sin[x]/11;
f2 = y*Cos[x]/10 + y*Cos[x]/11 - (1/E^y);
soln2 = Solve[{f1 == 0, f2 == 0}, {x, y}]

Solve::ifun: Inverse functions are being used by Solve, so some
solutions may not be found; use Reduce for complete solution
information. >>

{{x -> -Pi, y -> ProductLog[
         -(110/21)]}, {x -> Pi,
     y -> ProductLog[-(110/21)]},
   {x -> 0, y -> ProductLog[
         110/21]}}


Verifying

{f1 == 0, f2 == 0} /. soln2 // Simplify

{{True, True}, {True, True}, {True, True}}


Bob Hanlon


2013/3/3 Norman Polozka <normanmath at gmail.com>:
> f1=a*Sin[x]/l0 + b*Sin[x]/l1
> f2= y*Cos[x]/l0+ y*Cos[x]/l1 - (1/e^y)
> Solve[f1==0,f2==0,{x,y}]
>

• References:
  • How to find the roots of non linear equations
    • From: Norman Polozka <normanmath@gmail.com>